Personally I am alllrright with the general concept but I don't think it has been implemented all that well. Not all of these skills need feats and I would argue for cutting the list down.
Some are stupidly powerful and reliable in combat, menacing and silver-tongued I am looking at both of you.
Acrobatics - I am not a fan, too many combat implications and ignoring difficult terrain while not game destroying seems powerful and of regular use to an already powerful feat with a powerful ASI. It really doesn't need that ASI.
Animal handler - Stomps all over beast master rangers even further. (hey guys, one feat makes your class feature inept). I would argue that this one doesn't need an ASI either. It is niche, but powerful for the niche.
Arcanist - Would be better off with detect magic and identify, no ASI and no presti. I know it might seem powerful with two level 1 spells but the spells are still 1/pd and of very specific usage. Great for "magic scholars" who aren't wizards, or parties that don't have any arcane casters but still want to identify/detect magic.
Brawny, too powerful... doesn't need to be in the game imo. The double carry capacity has little or large implications depending on the DM running limits on how much a character can carry or not. But athletics expertise is too much to get with the ASI and as a feat.
Diplomat - Decent, arguably nothing special though. Generally getting advantage on subsequent interaction rolls with someone who is willing to talk to you politely for 1 minute isn't hugely important.
Empathic - I would rather this be called "insightful", but that aside... The bonus action is generally weak. It is humanoids only, an action and can fail.
Historian - Not a fan of the bonus feature. Thematically it seems irritating "oh you want to pick that lock, let me tell you about locks".
Investigator - No problems here, a bonus action search is situational but expertise in search is great and the ASI is great.
Medic - Weak, very weak. Medicine checks are already kinda moot with spare the dying and healers kits existing as well as the DC being a mere 10 normally. The bonus is really weak too and I would direct anyone to the Healer feat without a second thought.
Menacing - WAY too powerful, frightened is a powerful effect and the fact that this replaces an an attack means a fighter can spam it out without too much fear of failure due to expertise. The lack of real limits and the cost a measly attack is the problem here.
Naturalist - The spells seem odd, I would probably remove the ASI and bonuses while combing this and survivalist.
Perceptive - Doesn't need to be in the game. We have alert and observant already. The bonus ability is also kinda weak.
Performer - Very very weak and the bonus effect isn't something I would codify, I would argue this makes it seem like you cannot distract with a performance normally. Which means performance has even less uses that require a roll than normal.
Quick-Fingered - This is fine, again though probably drop the ASI... It doesn't have to appeal to everyone and rogues already get an equivalent. Appealing to niche builds with feats is fine.
Silver-Tongued - Good lord no, coupling this with menacing makes for a horrible fighter that WILL lock down single enemies in the most horrible of ways. Too many benefits from that bonus ability.
Stealthy - I am fine with the bonus ability, I am not a fan of giving out easy stealth expertise though as it really is powerful and if someone were to get expertise I would argue that the ASI is probably too much given the regular benefits they are getting anyway.
Survivalist - As I said before, I would remove the ASI and roll this into Naturalist while removing the spells. Survival RAW has a lot of uses aslong as the GM doesn't hand wave them all, but none of it is really combat or role breaking.
Theologian - -shrugs- I like detect evil and good, but am not a huge fan of thaumaturgy. Not because I think it is broken. I just don't think it fits.
I have mixed feelings here...
One open question is actually whether this 'double bonus' could be allowed to stack with Expertise. I don't think this is the RAI, but it might be an idea to preserve the Rogue's edge on skills, at the expense however of increasing that edge even more up.
RAW = No
RAI = No
"Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once."