Yeeseash. 21 pages of theorycrafting. I'm going to have to go out on a limb and say that this entire thread is kinda driving me crazy. I feel like everyone in here talking has no experience actually playing a blade singer.
Elves are my second favorite race after tieflings, so let me give you some feedback from my game involving a bladesinger.
8 Str, 16 Dex, 14 Con, 12 Wis, 16 Int, 10 Cha.
Bladesinger 7 (game end)
Warcaster Feat taken at level 4
Cantrips: Green Flame Blade, Booming Blade, Firebolt, Prestigitation, Shocking Grasp
Spells: Mage Armor, Shield, Charm Person, Absorb Elements, Detect Magic, Featherfall, Jump, Sleep, Invisibility, Misty Step, Mirror Image, Rope Trick, Spider Climb, Haste, Counterspell, Tiny Hut, Fire Shield.
Equipment: Two short swords (+1)*, Elven Chain, Boots of Elvenkind, Cloak of Elvenkind
*technically, this was a short sword moonblade that could split in two, but as the character was the first to use it, it had one rune, and thus was only good as a +1 weapon. Items traits are Elven (half weight), Symbol of Power (newly risen noble house in elf kingdom), Conscientious and Muttering (had the elven queen's personality and nitpicked. A lot).
So, here's how things worked. Early on, I would need to start off by casting Mage Armor every day. This was basically a given until I got the Elven Chain, which made the rest moot. That said? I don't think it really can be understated how much a wizard in mele needs to devote to using Shield spells. And I always tried to keep a Misty step in reserve. I had a terrible, terrible Str save. If I got restrianed, blown away, or knocked down, it could be the end. So, I needed to do my best to have backup spells to prevent that from happening. In time, Haste replaced any knock back, but I still kept Misty Step, because being Restrianed would be game over otherwise. In my opinion, if you don't have that much, you're FUBAR.
Most of the other low level spells known were devoted to exploration things. With low Strength, again, I needed spells to get past jumping challenges or climbing challenges that wouldn't risk the party. I had to be very careful with each spell, however. Each spell used meant that much of a chance that someone will actually hit me and end me. I needed to be very careful or I would die. Early levels are terrifying; you never know if you need to save your resources for now, or later. There was more than once the group grumbled about needing to help the weak elf somewhere. That said, my Dex and stealth training meant I could scout pretty well; not as good as a rogue, but good enough.
Once we hit level 4, things got easier. I got Warcaster, and had enough HP that I felt that I could survive a few lucky hits, and even if I did get hit, we had a Lore Bard with Healing Word to get me back up in an emergency. I could go up to the front line with confidence, survive with the help of Shield, and do decent damage. Warcaster let me equip two shortswords now - Bladesingers can't use shields, large weapons, but I could use two swords with ease. Most of the time anyways. The damage boost was slight, but I had my bonus action free most of the time, and I just couldn't see a reason to NOT do it. If someone tried to slip past me, I could use my Reaction to smack them with Booming Blade for a fair amount of damage, and they had to freeze, or take more damage. Now, this actually made me be a fairly good tank - slipping past just wasn't a viable answer anymore because of the significant damage ignoring me.
Level 5 came, and with it another game changing level. I got Haste. Double speed, another attack, +2 AC, advantage on Dex saves. This became my new "must keep up" friend. Level 6 was huge as well - Extra Attack. At this point, I had four attacks with Haste up (and lots of Concentration bonuses), 21 AC, shield spells prepped. I was hard to hit, and could do some very good damage.
At level 7, I took Fire Shield, and ended up using it for the final big boss battle. Turns out that when you're obviously on fire, the boss doesn't want to hit you anymore, and ends up attacking others, or relying on ranged attacks. Probably not the best move for a front line melee.
So, all in all, here's the results of the game. I was good at being a front line warrior. I had good AC, I had good damage output. I could scout decently well, and i saved a few low level spells to be able to make up for my lack of Strength. That said? With all this magic devoted to being melee and scouting? There was no chance in hell I could "wizard" like an Evoker could. Fireball? Fly? Ha. My magic was devoted to keeping me alive and not as dead weight when exploring a ruin. With my magic items and familiar, I made a decent scout as well. Again, not as good as a ranger or rogue, but I could hold my own. But that left absolutely no room to buff or protect other PCs.
Could I have just gone regular back row caster? Well, I suppose I could have, but I don't see the point. Without Shape Spell, I would have hit our clustered party with our evocations. I had to be careful on what I could cast. I could have been casting more spells like Hold Person, but on the other hand, the times when I had to retreat from the front lines, there tended to be more monsters slipping past the front lines to attack the back. Sure, bladesong protects against worst case scenarios, but those scenarios came up more often than if I had taken a different subclass. The idea that I can both melee it up as well as wizard it out is patiently ridiculous. Too much of my magic is spent making sure I stay relevant on the front lines and exploring - and even then, I don't think I was stronger than our Fighter who was using a halberd and Polearm Mastery. While I got my share of people with Warcaster, Polearm Mastery easily hit more often with their reaction that I got a chance to, and their bonus action attack added their Str in. I forgot what else they took - by end game, they were clearly better than I in raw damage dealing, and it would have likely gotten worse at level 8 unless I found some measure to take.
All in all, the I felt that my magic basically let me become a sort-of rogue or ranger. I basically had to devote all my resources to acting in the role of a melee rogue, and there was no time or energy for anything else. No, I was slightly worse, since I could just keep up with combat and exploring, but I couldn't do anything other than basic scouting - no survival, no thievery. A blast to play, but I felt like I occupied this niche of just barely keeping my head afloat, and things would fall apart at any time. I also felt like I had all but reached the extent of my abilities. From level 5 spells on through level 8 have not much to help out my character; it wasn't until level 9 that I had another spell to help me melee it out. That's problematic.