As for renting a vehicle, their requirements are usually a bit more stringent, things such as showing more documentation that I live where I say I live, they'll put a hold on a larger amount, but I've never been outright denied from renting a vehicle for having a debit card.
Of course, having said that, I was denied the last time I tried to rent a vehicle because they had started running credit checks and I have no credit. This despite the fact that I had been renting regularly from them for 7 or 8 years (a few times a year). Can you imagine thinking credit was a better proxy than your own records for determining someone's trustworthiness?
I've never liked rental car companies, they're all shitty in 1 way or another, but that for me was the last straw. It will be a cold day in hell before I ever rent a vehicle again.
Instead I went out and purchased a vehicle to do exactly what I was trying to do with the rental, I posted about it here: https://news.ycombinator.com/item?id=34099429#34114741
I don't think you get the same purchase protection with a debit card vs credit card.
As far as renting a vehicle, I've tried on a debit card and been denied. But that was quite a while ago so things could have changed, and I guess have as you have done it.