To be fair isn't flatpak encapsulating the user space portion of mesa, similar to any other chroot? In which case the apples-to-apples comparison would be shipping your own mesa alongside your app.
Which now has me wondering, is the common wisdom wrong? Could I actually statically link opengl if I went to enough trouble?
You need the exact right version that matches your actual graphics driver. So flatpak takes care of installing the matching drivers inside the sandbox.
Do you? I've often seen this repeated but at some point I tried a cutting edge chroot on an extremely (ie multiple years) out of date device and opengl seemed to work. It surprised me but then I don't know much about how mesa works under the hood.
To be fair the graphics APIs are provided as libraries with as much as possible done in userspace. Sandboxing that without any coupling at all would likely require either new kernel APIs or highly questionable virtual memory shenanigans.
You absolutely can't, since you need to link to the system libGL.so and friends for gpu acceleration, libva.so for video acceleration, and so on.