I think people should pick the best weapon they can afford and have proficiency in in the PHB, and use that, barring some overwhelming roleplay reason. For example, there is no point for a rogue who prefers melee combat to pick two daggers, or even one, over a pair of short swords. Even in the story, the characters would be aware that short swords are more damaging than a dagger, for the same reason they know a long sword is more damaging than a short sword, or a greatsword more damaging than a longsword. Their relative damage differences are definitely known in-character, otherwise you would have everyone picking the cheapest weapons from the weapon table instead of spending more on the best (and keeping the rest of the gold for armor or money to bribe the guards or whatever).
The claim that characters don't know weapons deal different damage is pure nonsense.
I would pick the best weapon that my character likes, if it's within one step from the best. For example, I like greataxes, but they are in fact mechanically inferior to greatswords. Especially with great weapon style tacked on. It bothers me, but not enough to play my dwarf paladin with anything but his grandfather's prized heirloom axe. Which I'm going to re-forge into a halberd at level 4

Because nothing will ever convince me, in the context of character ability, that polearms aren't simply above and beyond THE weapon to wield for melee str characters in 5th edition. Even if we found a +1 greataxe, I would still do way more damage and be more effective in combat wielding a polearm. It is the low hanging fruit of 5th edition, and silly not to reach for it. Ok bad pun
This is in a game where I'm a player.
When I DM, I have these houserules to re-balance the combat system in 5th edition:
1) All great weapons do 1d12 damage.
2) GWM style as I wrote before re-rolls 1s, 2s, 3s, as many times as they come up (but only for the weapon part of the damage, including crits, but not for smites or other riders). I'm considering making it damage advantage instead but need to double check the math first. So you always re-roll once and pick the higher number.
3) Duelist style gives +1 to damage when you go versatile so at least you don't deal less damage when you attack with both hands and go 1d10. 5.5 + 1 = 6.5, same as one-handed but with slightly better crits. Yes, this means you can combine the two fighting styles to get 8 DPR with a longsword used two-handed, same as a greatsword or greataxe. Not sure why you would do that, except if you're a paladin / fighter with a holy avenger. Holy avengers in my games are longswords only.
4) Polearm master doesn't add the extra attack and you can't use the -5 / +10 benefit of great weapon master on a reach weapon. Just doesn't fit thematically and is completely broken mechanically. These two fixes make polearms still very powerful, dual wielding the best to get multiple attacks, and great weapons deal the most damage in close range combat. There are also no magical polearms in my game, except if you enchant one yourself. Which is hard.
5) Rogues can deal sneak attacks only when not dual wielding, but get a +1 bonus to hit with daggers. This will make dual wielding short swords moot and a thing of the past. A rapier does 1d8 damage which is balanced I think against a +1 to hit, or vaguely so. I'm considering making their sneak attack damage the same as their weapon. So increasing from dagger to short sword to rapier makes sense as you gain more sneak attack dice. Will have to revisit this one with some solid math charts to see if the dagger's to-hit bonus needs to scale too, maybe from +1 to +3. Also, rapiers are expensive. Like, really expensive.
6) The weapon and armor costs will be completely changed to make the game more CRPG-like and give people a reason to save up their gold beyond just getting plate armor made. Great weapons are expensive, and if you want a long sword at level 1 legally, you are going to spend nearly all your starting gold on it. Name one instance of a PC in D&D ever picking up a non-magical weapon from the ground. You can afford everything you need with the default costs, life should never be so easy. Good quality steel weapons cost a lot. Maybe I'll make a table column for the iron prices (haha). At level 1, you pay the iron price because it's all you can afford. Iron weapons break on natural 1s. Good idea to keep a backup weapon.
7) Dual wielding is pretty dumb the way they did it. You should deal no ability modifier to damage on either hand, similar to Twin Strike in 4th edition. The twf style increases that to 1/2, and the feat makes it full. (including non light weapons and the +1 to AC and quick draw, as usual)
8) There are racial weapons that only those proficient in their use can benefit from. A Dwarven War Axe is just a Battle Axe with the Heavy property. Dwarves only teach other dwarves how to use them, after they have received some repute. The elven light blade is a greatword with finesse that can benefit from great weapon style. You need to spend time and do favors to get these proficiencies. It is rare that they teach these to outsiders or members of other races.