I don't think it is so much that availability doesn't show up on the kiosk, it is that things change a lot as people move or cancel FP+. We actually cancelled our A&E one day in the park around 11:00 a.m. because we had already seen them and wanted to get a different FP+ instead of seeing them again. I'm sure someone was thrilled to pick those up. Different things pop up all the time as people switch things around for one reason or another.
We have found the 4th FP+ can be useful at MK, and that you can even go beyond that. As far as the choices, it really depends on what is available. Usually, we take whatever we know has a long line; is available soonest; and is something we know we want to ride. Once we do that one, we try for a 5th, etc. BUT, If there was a FP+ we really wanted that was hard to get, like 7DMT or Peter Pan, and it popped up for a 4th, but was significantly later in the day- we would grab it and stick with that one. It is always worth checking on a 4th at MK even if it is mid to late afternoon as you can get some decent ones (unlike EPCOT or DHS where in our experience it is typically fairly useless to try to get a 4th at all).