Yes and no. Yes because the process would work in theory. No because the attestor (supposedly the OS?) wouldn’t attest your “headless proxy browser” as legitimate client.
Using a proxy would move the battlefield to how to trick the attestor. But realistically the whole thing will go down this route anyway. It’s another arms race. At the very end they’ll require cryptographic chips soldered into your device which make sure you’re not sideloading any software before running the OS, which would allow you to trick the legitimate attestor of the OS.