The brains are the issue, not the power source. If you operators can't see what is happening/control it, or the onboard autonomous mode glitches, you're gonna have a bad day.
https://en.wikipedia.org/wiki/Video_camera_tube
They're probably really inefficient and need bright lighting, low resolution, etc etc. But that kind of thing won't get zapped by radiation. It's glass and metal, maybe a few phosphors.
A CCD image sensor that’s lost 20% of its pixels could still be providing useful information, especially if you’re just trying to get the robot out. Other systems may inherently have issues long before that point.
Radiation hardening is tricky, there are tradeoffs involved and lots of reasons you might choose different points on the spectrum from "disposable robot" to "robot expected to have a long service life in a radiation environment."