...it should not be this complicated to do a simple cut. Why is it that developers HATE the right-click construct these days? Because of devices maybe? Anyway, once you have a part of a clip selected, all you should have to do is CUT, and the selected part of the clip is removed. You shouldn't have to do all that splitting and have to keep track of it.
For example, in Adobe Premier Pro, which can take up to 40 seconds to load on my laptop, all I have to do is mark in, mark out, and choose EXTRACT. And the selection's gone.
