Sorry but that does not justify his reaction at all. Not AT ALL so stop trying to make excuses.
And if he was so particular about how the car must be washed he can bloody well wash it himself cant he?
Yes she shouldn't have washed it that way but it's not the end of the world and he doesn't get to treat her like a little child that he can control with his tantrums and silent shoulder, no. He can fuck off to his apartment all he wants.
Yes she said she doesn't want him around because this isn't a one off, he sucks the air and joy out of every situation and the home and is usually off by himself in the room doing his thing ignoring his wife and kids. Don't try to spin it and act like OP went to the carwash to upset him because she doesn't want him around.
And on top of that she has to deal with a bunch of judgemental inlaws as well, what a wonderful bunch.
There's obviously no love in the marriage as things stand and with him repeatedly rejecting even attempts at marriage counselling and I've obvious OP is only hanging on because of her kids not because he has some amazing redeeming features.
You say she shouldn't chuck him out his home while ignoring the clear fact that HR threatened to move out and then change this mind when he didn't get his frosted reaction from OP aka she wasn't begging him please don't go!!!
Edit:
"How did I threaten him?. He said he'd be happier in an apartment alone and I said "go ahead and get an apartment alone if that's what you want" every other time he's said this I've backed down and said ",no please stay" "we are a family don't go" but why should I do that anymore.... if he wants to go just go ffs
Go to post"
I didn't even see OPs response before I posted and I knew immediately it was a controlling tactic, he threatens to leave and she has to beg him to stay and when It didn't work this time he quickly backtracked and then attacked her for wanting him to go. Classic emotional controlling tactic.