I like the ability to give any character weapon proficiencies of my choice. Changing it would cause my table to lose that ability with no replacement. Your suggested ability is just another "+ damage" ability which I find to be less interesting and less unique than the only feat that provides weapon proficiencies. If you want your "I want to do more damage … because" feat. You do just do something like "Reaper: character never misses and always does enough damage to kill enemies with one hit unless they are bosses." … because that's would end the relentless attempts to power creep damage and do nothing interesting with the feat.
I am not trying to be a jerk. I am just trying to make a point that being better at attacking and doing damage is not a "feat" of heroism and increasing those is not unique or interesting. Its just power creep and trivializing combat so you can kill things more than others kill things. Which is what you get from basic leveling in any class. If your trying to replace feat it needs to be something more than damage/to hit creep. Make it do something truly unique that makes the character stand out in a feat of skill others can't do.
I'm glad you added the part about not trying to be a jerk, because you did an incredible job of coming across that way in your first paragraph.
Now, I don't disagree with your point that feats are ideally about doing things other people cannot do. And you are right in damage being a very vanilla style feat. IF I had designed a new version of it (the version I am talking about is not mine, it belongs to the Dawnforged cast), I probably wouldn't have done anything like this.
But where I like this feat has nothing to do with your Reaper strawman. I like this feat for making sure my rogue who only uses daggers isn't falling behind the curve. I like this feat for making me still feel like a damage machine even when I use shields instead of greatswords. I use it not to raise max damage, but to bring lower damage up to the averages.
Now, you said you like the ability to give "any character" any weapon proficiency, and that your table would miss the loss of that ability. I'm honestly curious about who you use this feat for, because it has a deadly flaw in it per RAW that I do not like.
That is this. Who takes Weapon Master? People who primarily use weapons or who want to primarily use weapons would be the obvious choice.
For Barbarian, Fighter, Paladin, and Ranger this feat is completely useless in RAW, because they are already proficient with all weapons.
In the "eh, maybe?" camp, you have Clerics, Bards, Monks, and Rogues. Now, for Cleric and Bard, they have subclasses that give them all the weapons, so you would need to want a martial weapon, and have not picked a subclass that would have given you that weapon. For Rogues and Monks you would be looking into getting specific weapons that still work within their specialized combat styles. For Monk... I don't see why you would bother. All Monk weapons are already on their proficiency list and if you have a very specific weapon you are likely taking Kensai anyways to get it. Rogue is the only one I see benefiting from the RAW feat, since they could grab whips, Heavy Crossbows and Longbows, which all can use sneak attack and they are not already proficient with.
In the "why? Just Why?" camp I would put druids, sorcerers, warlocks and wizards. Maybe I could figure Warlock, but since pact of the blade already gives you everything you would need to have not chosen that, and then we have to discuss cantrips. Which really applies to Clerics and all the others as well. The point of getting a weapon prof is to use it in combat, you can already wield any weapon in the game, you just aren't accurate with it. And all of the casters except bard have way way better cantrips than the weapons they could get. With Xanathar's Druids get 1d10's or 1d12's in primal savagery, Clerics get 1d12's in Toll of the Dead, ect ect ect.
And final nail in the coffin. If you really want weapon proficiency gains (that somehow aren't related to doing more damage and doing better in combat) then why not use Xanathar's training system? It gives prof in tools and skills, zero reason you can't do the same thing for a weapon. And I'd argue you don't have to find and pay a teacher if a party member has the requisite prof, and with so many characters having martial weapons anyways, you are very likely to get what you need.
Example: Weapon Master... Choose one melee weapon without the two-handed property, while wielding only weapon(s) of that type (shortsword, longsword, hand axe, etc) and not wearing a shield, you may use your attack to manipulate lines of approach while maintaining your defense. Each time you make an attack against a target in 5ft, you may move to a another position within 5ft of that target without triggering attacks of opportunity from other enemies. The distance you can move can not exceed your speed.
So imagine a Ranger with two short swords or a fighter with a longsword and every time they attack the strike the opponent, parry the blade, and swivel in to a new position manipulating into or out of a group of enemies in order to escape being cornered or attack a target hiding behind guards. No additional damage or to hit but really useful and unique to all feats. Its use full and kind of an epic visual of a lone warrior manipulating his position through a crowd of multiple attacker who just can't seem to pin him down or hold him back... Kind of a master of his weapons...
So... this isn't a bad ability, but I'd say it is a really bad feat.
You pick one weapon, it must be a one-handed weapon, and you cannot be using a shield. So this feat is only appealing to dual-wielders off the bat, because heavy weapon users and sword and board are immediately disqualified.
Then, when you attack you get to move to a different space next to that enemy, without triggering AO's from nearby enemies. Meaning it is only useful when you are getting flanked or surrounded, and want to run from that spot. Which is a decently niche set of circumstances by itself.
But then I wonder, why not just use mobile? With mobile you attack a target and the target can't make AO's against you. So, in a 2 v 1 set up, it is indistinguishable (with the added benefit of working with any weapon or attack in the game). You may then call upon the 3 v 1 scenario, but since 1st feats are usually around 4th, you are likely hitting 5th level soon and all the martial classes get extra attack. Meaning 3 v 1 Mobile can do this as well.
And monks are rogues (who are the main skirmishers outside of ranger) can disengage as a bonus action, making all of this feat worthless for them. (Unless spending a point of ki is a big deal for the monk, but we are talking an entire feat here, that shouldn't be the comparison)
So, you've given a feat, called it "Weapon Master" that allows you to maybe have an advantage if you are wielding the right weapon in a 4 v 1 pile on, if you are skirmishing and need to relocate position. It's not very impressive.