Does Vegas or other NLE have 60 Hz notch filtering?
 

I know that Cool Edit does, and presumably other sound programs do also. But I'd like to do processing like this from within the NLE that I ultimately choose.

But of course it does. :-) And 50Hz filtering for PAL folk too.
But....building a preset for 60hz filtering might take as long as......60 seconds in any application that has an eq.

Vegas has a parametric EQ built in, so the answer is yes.

You can also use directX plugs-in like Waves.

Vegas can also allow you to do comb filtering to get rid of ground loop induced noise (but that may negatively impact your sound).