Could be that the igpu doesn't properly "hot reset" and that once the vm shuts down it gets stuck with the host.
This theory is easiest to test by checking if a shutdown, remove power, plug in power, startup of the proxmox host allows the existing vm to grab the igpu again.
If that works then a workaround could be to never shutdown the vm.
Proxmox
Proxmox VE is a complete, open-source server management platform for enterprise virtualization. It tightly integrates the KVM hypervisor and Linux Containers (LXC), software-defined storage and networking functionality, on a single platform. With the integrated web-based user interface you can manage VMs and containers, high availability for clusters, or the integrated disaster recovery tools with ease.
That was the only thing that I could find as well.
It dossn't seem to work on first boot, but I'm not sure that I have taken out the power fully.
It's really anoying that when it stoped it was only an update to the Neon VM that brought it all down. I cant remember any system updates at the same time.
I don't really want to buy a buy a gpu
I infrequently have the same troubles with my dgpu (Although running Debian with KVM and qemu instead of Proxmox) and the remove power and letting the machine be out of juice has been the only solution for my dgpu to reset properly in those situations.
I even press the power button a few times before plugging the machine in again, although I think that is completely unnecessary.
There are some "hot reset" scripts you can find when it comes to pci passthrough, but they did nothing for me.
So I powered off the server completely (removed power) and still no fix
If you're passing the iGPU to the VM then the host can't use it if the VM is on.
If your host supports SR IOV, then have a look at this repo: https://github.com/strongtz/i915-sriov-dkms
Installing this on Proxmox and your guest means the guest(s) and host to use the iGPU at the same time.
I'm not sure if it's in mainline somewhere but I had to install this DKMS to get it working.
Just realised your driving the TV with a VM. This probably isn't relevant then.