I know there's no crit on skill checks, but it's funny

There's no crits on skill checks in the book. Play how you want

Ok? The DC can be 15 in both scenarios.
While I agree that rules should be used more as guidelines, the last campaign I was in allowed crits on skill checks, and it my my lock (built partially around filling a "face" role, as everyone else min-maxed for combat) and I felt absolutely useless.
Our negative modifier int and cha barbarian player had a lot of lucky rolls, and thus was better than a cha based character with proficiencies in all the speaking skills...
I often felt left out of all aspects of the game really. Lock spell slots are limited, but is made up for by the short rests... If your party ever takes them. Bah.
- That sounds like a few problems and there should at least be a discussion with the group about expectations before future campaigns.
  But the inverse of a skill based build getting a lot of lucky combat rolls and outdoing the combat character is possible in the book. A good DM can and should mitigate skill crits but is kinda stuck on combat ones.
  Regardless, I'm sorry you had a crap time.
Crits on skill checks ruin the game because nothing becomes impossible, which ruins the story. You shouldn't e.g. be able to jump infinitely high by rolling Athletics until a 20 comes out, assuming you want your world to, you know, have prisons in it.
- Critical successes and failures can easily exist while still having the impossible be impossible, that's the DMs job
  Player rolls 20 on an intimidation check despite their roleplay being more awkward than intimidating? Critical success, because it's funny and entertaining usually to do so and it's theoretically possible anyway.
  Player rolls 20 on a strength check to lift a giant iron gate? You did a really good job of trying, but no, you're not strong enough to lift something 100x your size and weight
- In other words of what others have already said, a crit skill check isn't making the impossible possible, it's the best possible outcome you could hope for. Just like how a crit on a thing you can't hit is the best you could hope for. You don't instantly kill it, you just get a very good shot in.
  You don't convince the guard to let you go free, but maybe you manage to get him to believe you're inept enough that he can go to the other room and have a nap.
- That's not how crit skill checks work, so no wonder you don't like your version of them, your version does suck.
- It's role play, not roll play. Don't be a metagaming ass and just have fun

The rules as written, tell you not to follow the rules as written. Play the way everyone at your table wants to play. The basic rules are a guide for if you can't make up anything you like better.

The rules as written, tell you not to follow the rules as written.
So... We should follow the rules as written?
- Maybe!
BG3 handled it by saying what the hell, let the players have their crit success on a skill check.
And that is the greatest example I can think of where a dm follows the rules intimately and methodically but still welcomes the house rule of cool.

That's one of the many reasons 5e is complete trash.