Usually it's a management problem (like favoritism or something), so going to management is just going to put a target on your back.

Usually they've made a couple enemies. Just gang up with their enemies. If it's a group effort, there's only so much management can do at that point (depending on how much leverage they have). It's all dependent on the specific political situation.

This whole image of them being "well-liked" and socially invincible is just a play. Usually this type of person has made a shit ton of enemies. It may or may not be worthwhile depending on how shitty of a situation you're in, how bad the company's stack ranking and layoff quota is, how attached they are to you, etc

I think that’s a great reason to get into the psychology of bullshitters, or at least get really familiar with their tactics. You are going to have a lot of situations in your life when you can’t just flip a switch and get rid of them.

When you're working with a bullshitter and can't just leave, what use is there in performative caring? Your coworkers may have still cared, privately, but actively complaining about it, presuming boss man doesn't care anyway, is just pissing into the wind. You may as well accept that he's a bullshitter and get over it. Some people are like that and it doesn't help you to spend your time seething over it.

If you can get him fired then go for it. Otherwise, you're only harming yourself by letting yourself dwell on it. As for believing him; when my coworker starts rambling about the time the Navy seals came to him for help, I just say "wow man, crazy". I don't believe him, but neither do I confront it. Why bother? I'm not being paid to call out his dumb lies. This is probably what your other coworkers were thinking.