I just removed the bathroom in my '68 and I found that I had to remove everything to get the counter top out. The top (medicine chest) is riveted to a board on the top and either screwed or riveted behind, to the counter. AND mine was screwed beyond belief to the wall.
I kept having brain farts while doing this and ended up remembering that I didn't have to remove the faucet until the whole thing was out. Haven't tried that yet... I bought a cheap universal plumbers wrench for faucets etc. and it worked on those nasty copper fittings. But in the end I have removed all the plumbing... weird garden hose patches and floor rot...uugh the story is just beginning.