Of course, older people are not more important, and mil is not more important than OP, so if the mil sees the front seat as more important, that rules out her right to demand it.
It's never rude not to give up the front seat of your own car just because they want it, particularly if you are doing someone a favour and giving them a lift. You always have a perfect right to sit anywhere you like in your own car (or your spouse's car) and this freshly imagined "rule" that it's somehow expected isn't actually relevant as a) it's not a rule at all and b) even if it was expected isn't the same - at all - as required.
And even if it was rude not to offer - and it's not - the person having the favour done for them still shuts their mouth and is polite. Because they are having a favour done for them.
So that's the silly nonsense that it would be rude not to offer it on spec ruled out.
She's only in her 60s, has no health issues that would affect her sitting in the back and the back seats are perfectly comfortable and no different from the front. So that's any need ruled out.
The only reason OP needs for sitting in the front of her car (or her husband's car) is "I want to". So that's every other possible argument ruled out.
That just leaves us with MIL being an obnoxious, entitled, bullying, ill tempered, foul mouthed pos.
Wonderful that your husband stuck up for you. Guess she'd best be getting used to Ubers. She can probably sit in the front in an Uber if she demands it. Bet she doesn't. She just wants to make sure you understand she's further up on the pecking order than you are.
What a shame for her that, in fact, she is not.