Yes, hosting providers do run servers on desktop hardware. See for example Bisect's dedicated server options which all have 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.
Very large servers generally use server software like Folia which multithreads the game (like 2b2t) or split the server into many smaller servers (like Hypixel), which eliminates the single threaded performance issue but has some serious drawbacks.