This is why Intimidation is a floating skill at my table

Remember: RAW, the ability that gets used on a skill check is determined by the DM. STR is a perfectly valid ability for Intimidation.

On a related note, God I hate these skill systems, as an old-school DM. If you try to intimidate someone and it makes sense for it to work, it works.

Well you as a DM set the DC. If it makes sense to work then set it to 3 or something, or just make it free. But setting it to succeed on anything except for critical failure makes sense, since anyone can flub their grand moment.
- I also hate the DnD criticals. First, they don't apply to ability checks if you're playing by the book, so the point is moot here. Second, why is someone very skilled at something just as likely to crit as someone unskilled? Pathfinder 2E does it great where you need to be over/under the AC/DC by 10 or more for a crit. Someone very good at something will critically succeed more often with that skill than someone very bad, who will critically fail more often. In fact, someone particularly skilled may not even be able to critically fail a check that's trivial for them. The fact that a master still has a 1/20 chance to critically fail trivial things in the DnD rules isn't ideal.
- Also you can just give advantage if it's such an obviously threatening situation.
  That should easily make up for the lost + in Charisma modifier.
There are a lot of things I like about 5e, but charisma making you good at ALL forms of charisma simultaneously is one of my least favorite changes they made.
- I really dislike the 6 traditional stats for many reasons, and this is one of them.
  The chronicles of darkness games have a nicer stat system, in my opinion. It's 3x3. One axis is Power - Finesse - Resist, and the other is Physical - Mental - Social. They have names (strength, dexterity, stamina are the physical ones, for example), but this is the underlying concept.
  Demanding people's attention is Social Power. Being subtle is Social Finesse. And keeping cool is Social Resist. Now it's possible to make a character that is The Center of Attention who isn't subtle, or someone who cannot be spooked but also isn't very good at talking to people.
  If I was going to do some hacking to D&D, I would probably rip charisma out entirely. It's half-baked and its implementation introduces a lot of un-fun problems.
- Most of the times we decide spontaneously what ability to use for a certain skill. The fixed stat+skill is super annoying and breaks immersion.
  The wisdom 20 / int 8 Druid not being good in medicine? ... yeah maybe not good in school-medicine but knowing what herb can treat what illness is a thing of wisdom, not intelligence by default.
  Then, yes, Strenght for intimidation.
  Intelligence for deception - think of an elaborate network of pseudo-facts and weave them together in a complex way so the "opponent" is so overwhelmed that he just choses to believe you.
  And so on so on...
- least favourite changes the made
  Changes? 3e, 4e, and 5e all used it like that
If you try to intimidate someone and it makes sense for it to work, it works.
I can see making a CHA save for the victim, but (IMO) the DC should come from the situation, not the PC stats.

IMO players should feel like badasses. Rolls are for when they're doing something the hero in an action movie might fail at. This doesn't mean that the game shouldn't be challenging, but rather that the players should feel challenged by powerful foes, not by mooks. Thus I think the solution is simply not to require a roll when an ordinary person would have the skills to succeed with certainty. The barbarian would automatically succeed (at least on the surface level) in this situation.

I would only require a roll if:

The intimidator isn't obviously frightening or dangerous. The halfling bard would have to roll unless he's scarier-looking than the average halfling.
The intimidator is trying to be subtle. The barbarian would have to roll if he doesn't want to make a scene in public.
The intimidator is trying to get the target to stay intimidated even after the target is not in immediate danger. The barbarian would have to roll if he doesn't want the guard to run for help as soon as the barbarian is out of sight.
The target is unusually resistant to intimidation. The barbarian would have to roll if he's threatening a fanatic unafraid to die.
The target might tell a convincing lie. This is the fun case, because a failed intimidation roll will look like a successful roll until the barbarian walks right into a trap.

Can I play at your table?

I legitimately had someone try to argue to me that Kermit the Frog was more intimidating than King Shark.

Also, I like having every skill be floating and see what fun stuff people can come up with. I would recommend Intelligence (Acrobatics) if you're ever going to make a conspiracy and need to do some mental gymnastics.

I legitimately had someone try to argue to me that Kermit the Frog was more intimidating than King Shark
Guess it would depend on the situation. Renegotiating my contract with Kermit would be intimidating. Dude has been in the business for decades. I'd have to fight hard if I wanted top billing.
legitimately had someone try to argue to me that Kermit the Frog was more intimidating than King Shark.
They probably owed him money. I feel bad for them. I would not want to be indebted to that ruthless aquatic muppet.

In a world with magic a bard could be pretty intimidating.

intimidating the Ben Shapiro analogue NPC
"Talk, or I'll make out with you. And neither of us want that."

Pretty sure a normal goon would be intimidated by a barbarian. However, bigger physical strength is not always more intimidating: https://www.youtube.com/watch?v=9U1FVjRRlXY

Ok, here is my point: being able to crush a neck is strength. Being believable that you will crush your victim's neck is charisma. Of course you can torture someone. that is strength. threatening someone with torture is charisma. You have to be believable to be threatening. And that's charisma.

My beginner opinion (I'm in the middle of my first campaign ever) is...that doesn't make sense. Big muscles are a weapon and if someone is threatening me with a weapon, being scared has nothing to do with their charm.
Like would you be more threatened by the 140lb doorman with charm or the 240lb bouncer who just glares at you?
- I think that's because you're thinking about it with your body and your experience.
  Someone with a different body and different experiences might see that 240lb bouncer and think:
  Another guy they hired to be dumb muscle. I've dealt with his type before; wouldn't hurt a flea without permission. Would probably cry right after, too. But the little guy... his eyes are saying he'll do it. He'll enjoy watching the big guy crush my windpipe. And big guy? If the little guy tells him to, he won't hesitate.
  "H-hey, we're all friends here. T-Tell you what, I'll tell you what you want to know, and you can tell big guy here he's got nothing to worry about."
  If you take the right perspective, you can make almost any skill check make sense.
- Just have them roll a strength based intimidation check.
- How threatening is some big muscled guy with a high pitched voice?

This is why I allow players to swap skills/stats for checks, if it makes sense in the situation. In this case, they could replace the charisma stat with strength for the intimidation check.

Stats I understand, but if your character isn't trained in intimidation, they shouldn't really be able to improvise those intimidating phrases or scenes the player is role playing. That's why I prefer to roll first and rp after.
- I don’t think you need to go to bard college to demonstrate you can rip a man in half.
- but if your character isn't trained in intimidation,
  How does one "train in intimidation"? The 20 str dragonborn barbarian with a giant halberd on his back needs to write sentences to learn how to look scary? Will they be taught by their 8 str gnome bard with a flower in his hair how to show someone he means business?

“Let me try something.” My lizardfolk barbarian begins casually eating fingers directly from the prisoner’s hand.

“Wow, Hathis, you must really want that information.”

Up to the wrist now. “Information?”

I know it would be an oddity but intimidation has always been a saving throw. If the NPC can withstand the intimidation attempts feels a lot better then if a PC can successfully attempt to present an intimidation scenario

And what's the DC? That doesn't seem to affect the actual issue of how values are scaled.
- Same value it's just rather than sneezing part way through your intimidation or halving spinach in your teeth almost 50% of the time you attempt to intimidate, you just have people with mental fortitude not do be intimidated by your given scenario

....no, no, I totally get it, this makes perfect sense actually. The bard could hurt my feelings, which is way worse! I'll take the tall buff person choking me 👀

I’m suddenly reminded of the comedian who called out an audience member for wearing a shirt that said, “Don’t bully me, I’ll cum :(“