IMO, the Warlock is an exception to this because of the stupid Patron. There is a strong theme attached to the Warlock class, but it really is the only one.
I disagree. I think this is mostly a throwback to the older editions and while people tend to play to certain themes, I don't think they are engineered into the class for the most part.
The Bard and Sorcerer are both Charisma-based casters, but you can build almost any theme around either of them. Changing a sorcerer to Constitution doesn't really change the theme at all IMO, it just changes their spellcasting ability.
Literally the only thing that separates the three Arcane Full-Casters (Full-Caster equivalent for the Warlock) is theme. Sorcerers get their power innately, either being born with it or being magically altered sometime while alive. Warlocks get their power from making a pact with a powerful, magical entity/creature. Wizards get their power from studying arcane magic and figuring out how it works. They only exist as different classes because they have different themes. The different mechanics would not exist in the first place if the flavor text didn't exist.
Making sorcerers use CON instead of CHA for their spellcasting ability would at least cement the idea that Sorcerers are innate casters and Warlocks are bargainers that had to sign/make a contract/deal to get their magic. That wouldn't make the theme a ton different, but it would at least make the mechanics match the theme.
I don't think so. There is a difference in how they fight and some of them are easier to align to certain social aspects, but they are not exclusive. All three though get fearsome reputation ability and use it the same way.
The Rogue and Barbarian in particular can be very, very similar, I would argue even indistinguishable in terms of roleplay. You wouldn't know until combat started, and even after combat you might not know just from the description of it. You can easily make a Rogue that is a furious war machine and unless you listened to the mechanics with words like "Rage" or "Sneak Attack" there could be no thematic difference if that is what you wanted to do.
The wizard is a full caster and that is going to bring a different element, but it isn't going to be any different than another full caster like a Cleric or Sorcerer unless you make it so.
Barbarians don't get Cunning Action and have to focus on melee weapons, while rogues have to focus on Finesse/ranged weapons and have way less HP/ability to take a hit than Barbarians. Yes, they can be roleplayed similarly, but they are played in distinct ways and the mechanics influence how the characters are roleplayed.
You quite literally said "mechanics don't influence roleplaying" and moved the goalposts to "well, except for spellcasters, because that's different".
1) I think the class should not be the character identity. The background, backstory and potentially race should are your primary theme. IMO you should pick a class where the mechanics will work with those things and use the class features, INCLUDING feats to make that.
2) Remember feats are class features, they are in the class tables and are part of the feat chassis. Unless you play a variant human or custom lineage, your class is the only way to get a feat.
1) It is not your whole character identity, but it is your core mechanical identity. Race, background, feats, that's all just extras that you add to your character idea to build onto it. There's a reason classes are the most mechanics-heavy of the 3 defining character-building options (race, class, background).
2) No, they're not. ASIs are class features. Feats aren't. They're optional, ASIs aren't. Variant humans are also optional, as are Custom Lineages, as is made quite clear in their text. Feats are optional, classes aren't (yes, DMs can exclude certain classes, but the class system as a whole is a core part of 5e that cannot be taken out without completely changing the game, while feats can).
There is nothing that prevents bladesingers from using the GWM feat, using shields or using medium/heavy armor or using any of those weapons.
Yes, "something" does. Read the Bladesong feature, please. It cannot be activated while you're wearing medium/heavy armor or a shield, and you cannot wield two-handed weapons (or versatile weapons with two-hands) while using Bladesong. They are quite literally incompatible RAW. That's one of my major complaints with the "just play a bladesinger!" argument, because the Bladesinger heavily restricts the possible themes that a true, complete Arcane Gish class/subclass should be able to use (medium/heavy armor, shields, two-handed weapons, etc).
Take mountain Dwarf, trade one of your weapons for a maul, choose bladesinger subclass and at 4th level choose GWM. You are now a bladesinger that can wield a Maul using BB/GFB with GWM. Pick a different race without armoer and weapons and you can still do it all, it just takes more time to come online. You are now a bladesinger that can wield a Maul using BB/GFB with GWM. Pick a different race without armoer and weapons and you can still do it all, it just takes more time to come online.
Now if you want shields too you will need another feat, but if you are wielding a maul you probably don't want that. Take medium armored feat at 4th level instead if you want to sword and board.
Assuming a 14 Dex with shield spell she has a 22AC while swinging a maul, when she hits 6th level she gets bladesinger extra attack combining an attack and magic as one action and can do it with a maul. There are other abilities she can't use, but if this is the character you want to build those other abilities she can't use are not really important anyway.
If that is what you want to build, then yes you should take feats to do it. As I noted above feats are part of the wizard chassis, they are class abilities, you get them from the wizard class. If you want to play an Arcane GISH and Ranger suits you better, then play a Ranger and take magic initiate as a Ranger feat to pick up booming blade and green flame blade. What I don't get is the argument that you should not have to use a feat when other classes do.
I addressed all of this above. Again, read Bladesong to see why all of this is wrong.
Tavern brawler is a fine feat for a strength-based bladesinger if that is what you want. It gives you proficiency in improvised weapons which you can use to make an improvised weapon attack with vials of acid, poision, oil and holy water and it gives you a bonus-action grapple after that (which can be enhanced by spells). You can attack with a vial of oil and follow up with Green-flame blade and get the extra fire damage from the oil on the same turn while also getting a free grapple. You can also use holy water or acid to stop many opponents from regenerating in the same turn you hit them with your GFB (again in addition to grappling them). With the free grapple you can position enemies so your 2nd enemy GFB damage lands more often (in addition to all the other things you can do with a grappled creature). If you add the grappler feat to this you can potentially get advantage on all of your attacks after the first improvised weapon strike, which a huge damage boost. This lasts until the enemy uses an action to TRY to break it. I am not saying that is what you should play, and you will need to either run a relatively low intelligence or dump social skills completely to do it well but it is totally viable as a play style
Tavern Brawler a) doesn't turn a Bladesinger into the Arcane Gish that we want, and b) even if it did, it would take a feat to do so. Paladins don't require feats to be paladins, so neither should Stabnerds.
I agree on the Warlock, and I have not played an Artificer, but I disagree on both the EK and the Bladesinger and if you find this to be the case I think it is because you didn't build your character out to do what you actually wanted to do and instead built to some predefined stereotype. You can play the character you claim you want to play with either of these.
I will say a Bladesinger can do GISH better than an EK, and honestly better than any other build, primarily because their extra attack feature is better and they get more spells, but there is nothing saying you need to take fireball and hypnotic pattern. Take spells that identify with the GISH character you want to be.
Thank you so very much for telling me that I didn't think about my character build enough. That truly means a lot.
Please do mind the causticity, though I do feel that it's warranted. It is quite rude to say "if you aren't satisfied with the options available to you, it's because you did it wrong!", and I very much do not appreciate that and will ask you to never do that again to me or anyone else in the future.
I detailed why Bladesingers don't work (and explained why they didn't work how you thought they did). I detailed why Eldritch Knights don't work (largely because of them being restricted to only 4th level spells, them automatically gaining cantrips, even though Rangers and Paladins don't, them being restricted to basically just two schools of magic from the Wizard spell list, and they can't prepare spells the way that a true practitioner of merging spell and sword should be able to (in my mind, anyway), and they're not at all good at merging spells with swords in any form (and no, just spamming Shadow-Blade and GFB/BB doesn't count).
I can't play a mediumly-armored Elf that puts a lightning bolt into a scimitar that is released when the spell hits, or a Dwarf with a dwarven thrower that releases a fireball when it hits a giant in the face, or anything else that screams "merging arcane spell and weapon with each other".