How can I run a VM on the server where Proxmox is installed, rather than in a client web browser?
All VMs you start are running on Proxmox.
Usually
none of them uses the physical display at this machine. This is what you want, right?
The pure fact that you are asking this make me hesitate to recommend something, but here are some options:
PVE is based on an unrestricted/unlimited Debian Bookworm. You can install any additional software you want. This includes a Graphical Desktop Environment. Note that in the same moment you do that you are leaving secure terrain - those additions are simply not
officially supported by Proxmox.
An official hint is here:
https://pve.proxmox.com/wiki/Developer_Workstations_with_Proxmox_VE_and_X11
The result is a Desktop Environment. You need to login as a user. Open a local Webbrowser. Login to PVE. Start a pre-prepared VM. Connect to it (possibly using Spice). Make it fullscreen. Select the right resolution inside the VM. ... ...
(Older statement:
https://forum.proxmox.com/threads/using-vms-on-the-same-host-machine.153004/post-694760)
A
different approach is to pass-through a graphics card to a dedicated VM and make it autostart. The expected result is that PVE boots up, starts that VM and after some short amount of time you get the output of a VM directly on the physical screen. Note that this might be difficult to setup - first you have to wade through a
bunch of problems as not every hardware will do what you want it to. Disclaimer: I've never tried this.
Good luck. And more important: have fun!