Yes, hosting providers do run servers on desktop hardware. See for example Bisect's dedicated server options which are all desktop Ryzen processors.

Smaller Minecraft servers can run just fine on standard server hardware, and I'm sure that's what most hosting providers use for their lower tier server plans, since you can cram dozens of Minecraft servers onto one machine. But the main limitation for larger servers is single-threaded performance, which desktop CPUs are just far better at.