As for the weapons table; first, the omissions: 1d12 and 2d6 piercing weapon, 1d8 finesse slashing weapon, etc. Then the handaxe, should be 1d4 damage, also, the greatclub should be heavy and 1d10 damage. The quarterstaff being versatile is stupid. The pike and trident are a mess (pointless). Oh, and the net, you always have disadvantage when attacking with it, marvellous stuff.
I'm pretty sure that they went with a commonsense approach to iconic weapons rather than a point-buy system or table-filling exercise. Some weapons are in there because monsters use them rather than because it was intended to be a balanced player choice.
For example: you have the pike as the larger piercing weapon. What weapon would you put for 1d12 or 2d6 piercing that isn't just a variant of it.
Likewise the general perception of the rapier is sort of justifiable as a d8 Finesse, but finding a slashing weapon on par with a longsword that would do the same damage whether it was Bilbo Baggins or Bruce Lee wielding it is tricky.
Have to agree about the quarterstaff though.
That's exactly it. There is no reason why rogues or bards should have the potential to be better at skills than the other classes. There is also no justification for them to be "skill monkeys". To me, it is a lazy way to make them stand out in an unreasonable way. Both these classes are already the only ones who begin with more than two skills for their class. As it is, with a flat expertise bonus (say +2) they still have the potential to be better, but not a full +6 better as in RAW.
The justification for the rogue to be a skill monkey is part history and part balance. The rogue has always been the mundane skill expert.
Their ability to use skills better than other classes is to make that capability reliable enough to compete with outright magical class abilities such as spellcasting.
The Rogue has the potential to get a higher result on a climb check than other classes? The Monk can run up the wall. The Wizard can cast Levitate, or Fly, or Spider Climb.
The Rogue can reliably pull off Persuasion and Insight? Wizard can cast Suggestion, Detect Thoughts etc.
A specialised Rogue has a better chance of recalling the history and effects of a magic item than a wizard does? Wizard can just cast Identify as a ritual, or even legend lore.
Skills do not run out of uses (while spells may be unlikely to as well in most adventuring days, utility spell use can detract from combat capability.)
Spells are often either automatic success and/or can do things that skills cannot.
Other classes, even ones that also get spells can use skills as well, however, which means that to make the Rogue stand out, they have to make them better at using their skills. Better to an extent that they can pull off extreme things reliably enough that the party doesn't just let the spellcaster do it.
Hence: Expertise in its current form.
Sure, making the skills they are good at useful in other ways is great. Something that would make sense for a rogue/bard but not other classes.
The difficult thing is just about anything I've thought of along these lines makes just as much sense of other classes, too.
That bit I've bolded right there. That is the issue that you're always going to have with nonmagical classes like Rogues and Fighters being able to do cool and unique stuff. No supernatural element or other special endowment (Hur hur) means that technically, there shouldn't be anything that they can do that other classes can't:
"My wizard is a sage, just like the rogue is. Why can't he get as high on an arcana check?"
"My paladin is in combat just as much as the fighter. Why can't he get four attacks?"
. . . and so on.
You either allow the fighter and rogue to do mundane stuff like swinging weapons or sneaking around better than other classes, or you need to give them something that no-one else can do. But since they are a mundane class, there is no reason that other classes shouldn't be able to do that. . . . and so on.
Overall, consider the attack roll, ACs (IIRC), and bounded accuracy:
At Tier 1, the average opponent AC is 13 or so. Most characters will have a +4 or +5 to their attack roll, so need a 8 or 9.
At Tier 2, average opponent AC jumps to 15ish. Fine, most characters have gained a +2 or +3 to their attack roll, so still need about 8 or 9, give or take a point.
At Tier 3, AC rises to about 18, but characters keep pace via proficiency increase and ASI typically. They might need a bit more than 8 or 9, but only a point maybe in most cases.
At Tier 4, AC average is close to 21. Assuming max ability, characters are looking at +11. So, 9 or likely a 10. Not far from the 8 or 9 they needed back at level 1.
That is because 5e is specifically designed to keep attacks to have a reasonable chance of hitting at all levels, with HP being the scaling factor in tiers. Hence it scales differently to skills where the DC is the scaling factor.
Save DCs vs save bonuses along the tiers would probably be a valid comparison.
Now, look at DCs for checks. The max (in theory) is supposed to be 30 as I understand it. Which works fine and in a similar fashion to the attack roll versus ACs. At high levels, PCs have about a 50/50 chance or better to accomplish a Hard (DC 20) task. Personally, I am fine with that. I don't want Hard to become "Mundane". But...
You throw expertise as number boost and now a Hard task becomes easy. Giving bards and rogues a boost equal to proficiency at lower levels is nice and not overwhelming, but as levels increase and the skill jumps two points each time instead of one, it becomes more of an issue.
I am of the opinion that a rogue approaching epic levels who is specialised in that skill should be able to regularly pull off feats that the average peasant would probably regard as nearly impossible.
We're talking about a level where
Wish spells become a factor in encounter design. Having a party member able to reliably sneak past most monsters they encounter, as long as there is cover and only mundane senses at work does not seem game-breaking, or even comparatively problematic.