Whose to say you can't do that with the Shield? The only issue is that it doesn't do it at a great price point of $300. Anyway, the Shield has a HDMI Out, USB and Bluetooth. So, if you really wanted to, you could always plug it into your TV and use a separate controller with it.
I've mentioned this before, but I think NVIDIA would be much better off creating a cheap set-top version of the Shield for ~$100. This allows you to stay in the PC ecosystem even if you don't want to sit at your desk anymore.
I use Splashtop quite a bit for remotely connecting into my desktop, and there's one huge barrier compared to the Shield: Splashtop doesn't support full-screen gaming!
Isn't the Vita rather limited in what homebrew it can run? I believe I read that it can only run PSP homebrew, but I don't really keep up with the Vita hacking scene much.