Privacy.com issues you a card that is coded as a prepaid debit card. Most payment providers expose this to merchants, which means that if they choose to decline prepaid debit cards for other reasons you'll get caught in the mix.

What's the reasoning behind rejecting prepaid debit cards, if there are sufficient funds available?

Sometimes credit cards are used kind of like collateral... when you stay at a hotel, for example... you give the hotel a credit card for 'incidentals', and it is basically being used as a guarantee that you will pay for things you are going to use. Car rentals are the same.

In a sense, it is your credit card limit is basically your credit card company saying "yeah, this person is good for x amount of money". That isn't much for a prepaid card.

Because then they could no longer collect money from people that forget that they're being automatically charged every month.

Credit cards are sometimes used as a proxy for identity (via billing address) or age verification (most under-13s don't have ready access to one).

Maybe the cards are being loaded with illicitly acquired funds

That's a valid risk, but then why single out prepaid cards for this given the rampancy of credit card fraud and theft?

Edit:

An interesting answer elsewhere in the thread: https://news.ycombinator.com/item?id=20465459

Yes I've noticed it too. I think they tell them this because Maybe it's to prevent freeloaders from creating multiple audible, Netflix, etc trial accounts?

Unfortunately it also defeats the legit purposes too.