The repo maintainers for Sunshine only build for the LTS versions of Ubuntu. Which is fair enough.
I run Kubuntu 25.04 on my desktop at time of writing, and the LTS build versions of Sunshine would not work. It expected certain libraries to be a certain version and it just wouldn't start. The AppImage and Flatpak versions had their own problems. Documentation on building and its challenges were decently sparse but some sharp Google-fu despite Google's lobotomization enlightened me to the solutions.
I write my experiences here in case it helps anyone who needs it.
First, I created an Ubuntu Server 25.04 VM on my Proxmox server. That's just me, you don't have to do that, I'm just insane.