Sure, you can lay new tracks over a live track all you want. Give it a try and see how you like it. If it comes out good, it's a keeper.
If the tracking room is tiny and/or has poor acoustics, try adding short reverb to some of the tracks to give them more of a big room sound. Be careful though, it's easy to over-do it.
Lately I have been experimenting more with reverb. I route all of the drum mic tracks into one stereo subgroup. Then I apply verb to the whole kit on that track. I solo the subgroup and then go through all of the presets on all of my reverb plug-ins to find the sweetest sounding one. Then I tweak it to my own taste to get it sounding real. It really gives the kit more life.
I sometimes put a "slight" amount of verb on the final mix as well (on the master L & R tracks.) Some of the verbs sound somewhat ugly. You really need to find and tweak just the right one for each song. And some songs may not work with it at all. You'll know when it is right and when it is not. Let the songs tell you.