(Although I admit I lockpicked the BoS door and went to the glow when I had PA for the rad res)
I just did that part and didn't bother curing the radiation and it was fine. I got back to the BoS base with zero radiation again. I'm not even sure what radiation is meant to do in Fallout. I remember always being wary of it but I'm not sure why now.
But as for your other (non-quoted) point (and slightly spoilerish stuff follows), I agree that doing it that quickly would ruin it - at least the first time. It's just nice to know that, in a believable sort of way, the world is all there from the start and you don't have to jump through about 60 hours of forced narrative just to make the final boss appear so you can kill him. I haven't done the Military Base yet, but I do know that with a high enough speech you can just talk The Master out of his plan, so if you go straight to his location, it'll be a very quick process indeed. You don't need to get the water chip if you defeat The Master and complete the Military Base in time.