underneath- Having no protection on your paint leaves it vulnerable to all sorts of things, from UV radiation to bug guts and bird bombs.
Yeah, after claying a good time to polish since the paint is clean. But you can polish whenever you need to. *I* only do it when I have marring that I can't live with. For general "paint cleaning", I use clay more than anything else (though some mild paint cleaners certainly have their place).
Some of us clay without it seeming to strip our protection. I just clayed Accumulatorette's A8 (tar and bugs) and its protection (3M paste wax) is just fine. I spot-clay our vehicles with every wash, and it doesn't seem to strip any protection. But then I clay *VERY* carefully and gently, trying to pull the contaminants out of the wax/sealant. No "heavy rubbing".
Since I only reapply the LSP to our drivers when necessary (usually only a few times a year, sometimes ONCE a year with Klasse), I'd surely notice if the clay stripped my protection. So it CAN be done that way. But if you clay too aggressively, without adequate lube, then yeah, it'll remove your wax or sealant. And if you have contaminants UNDER your LSP, then you have to go through (remove) the LSP to get to them.