I’ve heard good things about Proxmox, but I have no direct experience with it. That would be a separate box that manages the VMs and everything, and it has a remote GUI option (webpage I think?).
If you want something on an existing box, just use KVM directly, or a simple frontend like GNOME boxes. I don’t know about remote configuration, but once it’s set up, do you really need to check in on things remotely? KVM will do hardware acceleration (definitely CPU acceleration, GPU if you configure it properly), and it has no GUI by default.