As it was going to host (among other things) my mail server, I could not use my primary (self-hosted) email address - because that's asking for trouble if I ever run into issues. It seems using Protonmail triggered something on their side.
They asked me for a copy of my ID card, which I happily provided. But they still refused my account, without explanation. Oh well, their loss. I've been at OVH now for a few years without any issues. OVH's product is definitely worse, but at least they'll actually let you use them.