I think this is all a bit off, frankly.
I recently appraised of the phrase "special pleading", and it seems like it applies really, really well to what you're posting here - specifically you have a massive double-standard in which a caster firing off a cantrip or using a moderately effective weapon is "zzzzzzzzzzzzz" and "boring", but apparently a non-caster making a similar contribution, only with possibly a higher damage number is somehow having a great time.
Most other stuff seems a bit dodgy too. You seemingly ignore the utility of utility cantrips, you seem to think casting utility spells is merely some sort of onerous chore, preventing you from casting REAL spells (i.e. combat spells), rather than something that most people think is awesome, you ignore most of the class/subclass features of casters (seemingly), and most strangely of all, you seem to wildly overestimating how long 5E combats against normal (i.e. not legendary) opponents last, which seems, from what I've gathered, to be more like 1-4 rounds, rather than 10-15. You complain that "3 spells per encounter is unobtainable!", but how many encounters will even last 3 rounds? If they do, how many will have those rounds beyond 3 as anything but "mop up"? Especially if you open with some sort of nasty spell. Not many, I'd imagine.
With Teleport, it's a 7th level spell, but perhaps you mean Teleportation Circle, the 5th level one? I seriously doubt any normal party will have an expectation of you casting that twice a day. If they do, it's a huge deal, and they will certainly think it's important, as, presumably, will you, rather than seeing it as a "waste" or "zzzzzzzzzzz".
You dismiss Burning Hands, I note, but it does 3d6 damage (so avg. 10.5) in a 15' cone, which, even at 10th level, is likely to be a spell worth casting on grouped enemies. You keep talking about "10HD" and so on - but PCs at 10th level aren't consistently facing 10HD monsters - they'll often be facing larger numbers of much lower HD monsters.
And on top of this, you're basing all your assumptions on a 6-8 medium or hard COMBAT encounter a day structure, which, frankly, seems very unlikely to be what we actually see in the DMG, and I believe is not reflective of current WotC 5E adventures (I don't own any, so people who do feel free to correct me, but the impression I have got from threads, reviews, and so on is that whilst you may be looking at that number of encounters a day, not all are combat, and not all are medium+).
Err, yes I am basing it off 6-8 medium or hard combat encounters a day. From the "Building COMBAT Encounters" section of the DMG:
Assuming typical adventuring conditions and average luck, most adventuring parties can handle about six to eight medium or hard encounters in a day. If the adventure has more easy encounters, then the party can get through more; if it has more deadly encounters, they can handle fewer.
If the section on Building Combat Encounters states 6-8 medium or hard encounters in the DMG, I assume that this is what they are talking about. So yes, my discussion is only on combat (and not on using cantrips out of combat). I also assume that the guideline means half medium and half hard encounters each day.
At first level, a medium encounter is 50 XP per PC. A hard encounter is 75 XP per PC.
So, a 5 PC group should be able to handle 3 to 4 encounters of 250 XP and 3 to 4 encounters of 375 XP.
That's 3 to 4 encounters with 5 Goblins (one hit foes for most PCs, but nastier than on paper if they are not out in the open and can use their abilities for hit and hide tactics). Alternatively, that could be 5 Giant Poisonous Snakes (11 HP, AC 14, +6 to hit, D4+4 plus 3D10 failed save, half made save), sometimes one round (2 handed hits, or 2 wpn, both hits) foes for the martial types. Ditto for a lot of other 50 XP foes like Giant Wolf Spider.
And that's 3 to 4 encounters with 2 Hobgoblins (Hobgoblins can sometimes hit hard for an average of 12 damage) and 3 Goblins (or 3 Hobgoblins and 1 Goblin), or 3 Gnolls (Gnolls do not hit as hard as Hobtgoblins, but they have twice as many hit points at 22, so they are more durable, 2 rounds of martial successful hits each, 4 or 5 cantrips to take one down) and 1 Goblin.
Btw, 3 Cocktrices (27 hit points each) is a 300 XP fight where one failed save restrains and two failed saves petrifies a PC for 24 hours. Granted, they are easy to hit and have a hard chance of hitting back, but some unlucky rolls and this one (semi-medium, semi-hard) encounter by itself prevents even one more encounter from occurring. In this particular case, sure, a wizard could do sleep or burning hands, but 3 small little creatures, he might not know early on that he should be doing this.
Sure, we are talking foes that have a 35% to 45% chance of hitting many front line PCs (and higher against squishier PCs), but even so, that's maybe ~2 hits the first round, ~1 hit the second round, and 0 hits the last round for medium encounters and one more hit for hard encounters (since the hard encounter monsters have more hit points, they last longer). 3 hits per encounter * 3 - 4 encounters * 5 points of damage per hit plus 4 hits per encounter * 3 - 4 encounters * 8 points of damage per hit (tougher monsters either get more attacks per round, or hit harder) = 141 to 188 hit points / 5 PCs = 28 to 38 hit points per PC. Every PC goes down about 3 to 4 times (hit point dependent) and every PC gets healed back up full.
PCs just do not have these kind of resources. Most groups can heal back up ~50% of total group hit points (maybe 25) via HD in short rests plus maybe 15 to 20 via a Cleric, 40 by a Bard (15 plus 9D6, assuming a Cleric or Druid in the party healing only during short rests) plus maybe a Fighter 25 (5D10, 4 short rests). That's 105 to 110 assuming that they have two healers, one of them being a bard and that they have a fighter. They cannot even handle 6-8 medium encounters (assuming the DM plays the NPCs reasonably).
141 to 188 damage. 105-110 healing (80-85 without a Bard or a Fighter, 55-60 with neither a Bard or Fighter). Sounds like a TPK. Especially since PCs have to survive the damage in order to get the Bard healing.
This assumes that there are two healers in the group who never use their spells for anything but Cure Wounds.
This assumes encounters where there are not more hits.
With an NPC ambush, these numbers blow up more. With the squishy PCs getting attacked during a few encounters, these numbers blow up more. With lucky NPC criticals, these number blow up more.
Granted, with a PC ambush, these numbers decrease. With lucky novas or PC criticals, these numbers decrease.
Sure, PCs get more hit points and abilities at higher levels. We'll have to see how that works. But if it seems off at level one, then it might be off at higher levels too.
Regardless, your comments on 6-8 encounters do not seem supportable. It's WotC's guideline. Although I think it is unsupportable based off party resources, I do think that it is the baseline for where discussion has to start.
Also, on the Burning Hands question, I'm not seeing it. Burning Hands is 15 feet away, 15 feet wide at the end. If 4E illustrated anything, it's typically difficult to get more than 2 NPCs bunched up in a blast. 4E solved it by having a lot of multisquare pushes and slides. 5E does not have that. On a grid, 4E blasts were 9 squares. On a grid, 5E cones are 6 squares (i.e. 1 square 5 feet away, 2 squares 10 feet away, 3 squares 15 feet away). So the odds of 3 NPCs being in those 6 squares tends to be limited. DMs might group enemies around a PC and swarm him, but only Evokers can really use Burning Hands well in that situation and get 3 foes. Other casters would typically get 2 foes on average (because they would rarely use it on 1 or 3 and would save it for 2).
2 NPCs. 40% chance to save each with +0 to save. 36% chance of 20 damage, 48% chance of 15 damage, 16% chance of 10 damage. The most likely outcome is one saves and one fails basically half of the time. These odds shift against foes with +1 or more to their save (like Goblins 25/50/25 and Hobgoblins 30/50/20). So, one dead and one wounded for Goblins. Compared to the martial types who often have one dead in the same timeframe.
Sorry, not seeing where Burning Hands is all that great. It's about 50% more damage than a non-shield melee type can do whereas Firebolt is about 50% of the damage a non-shield melee type can do. Burning Hands gets worse as the caster levels up. Btw, I suspect that a lot of players of melee PCs will forego shields. 5E is a game where offense trumps defense. 30% to 40% more damage every hit trumps 10% better change to get missed.
With regard to 10 HD creatures, I agree with you. A single Hill Giant is 10 HD. It's considered a hard encounter for level 10 PCs. It would be lucky to survive two full rounds with 10 PC turns (14 to 16 attacks class depending) against it. It might hit the PCs twice for a total of 40 points of damage against PCs that have 50 to 100 hit points each. It's more of an annoyance and slight party resource hog than a hard encounter. 4 Griffons with the same XP, on the other hand, do half of the Hill Giant's damage per round, but with more than twice as many hit points is an encounter that will do more. They'll hit maybe 10 times for about 100 damage and it will probably take at least 3 rounds and maybe 4 to take them out.
Btw, you misinterpreted me. I think that the vast majority of 5E encounters will be 2-4 rounds. One round will very rarely occur. 3 rounds should be the norm (sometimes 2, sometimes 4). As an example, if the PCs surprise their foes in a surprise round, then all of the PCs attack. Some of those PCs will miss and because it is a surprise round, most casters will be saving spells until they see the outcome. Typically, some foes will die, some will still be alive for round one. I seriously doubt that one round will be frequent. An occasional large AoE could result in a single round once in a while.
I'll also point out that your "special pleading" comment was not supported by the rest of that paragraph. Show me where I ignored something pertinent concerning PCs who do 50% of monster damage per hit vs. PCs who do 25% of monster damage per hit and how for some people, doing significantly less damage might be considered boring. If pinging the foes is exciting to you, great. It's not for some other people and not a double standard. The vast majority of players do not play the game to be partially ineffective round after round, encounter after encounter. You might think that is enjoyable, but there are many people who might get bored being the 5th wheel a lot (at first level in a 6-8 encounter day, 3-5 or 4-6 of those encounters without a non-cantrip spell, class depending).
My personal take is that most adventuring will be 4 - 5 medium or hard encounters at tier one and that will climb by one encounter per tier as PCs get more resources, especially (heaven forbid) magic items.
Btw, I am also concerned about the fact that 2 of the 3 major saves (2 of the 3 minor saves) at high level do not have a proficiency bonus for most PCs. High level fights, not even including legendary, might be pretty nasty when a PC fails a save 4 rounds in a row, especially if that is a PC that does most of the party buffs or dispels.