Nope, you still suffer from radiation poisoning, you just regenerate from damage while doing so. And radiation doesn’t kill you through damage. (I mean, technically it DOES, in the form of a single unblockable 500 damage hit to all locations when you hit the lethal threshold, but radiogenic isn’t going to help with that)
Playing a radioactive character without any rad protection is possible, but it requires you to keep moving constantly, as radiation rapidly builds up whenever you stay in one spot for more than half a day or so. You could alternatively just live in an ANBC suit, and give them to your companions. This really shouldn’t work, logically, but it does.
I’d definitely avoid taking Elfa without Repair Nanobots though. Not only do you continuously lose a tiny amount of health, but you no longer need any sleep whatsoever, and you only regenerate health naturally while sleeping. Interestingly, Radiogenic COULD substitute for the nanobots, but you’d have to deal with being at least slightly irradiated for it to work.