I never used the Push and Start feature. Maybe it does not work if there is no separate Clearance frequency? I don't know.
At airports such as LIMZ which only have a Tower frequency the push and start feature even if selected doesn't operate. That's pretty much correct actually. An airport with just a single Tower frequency is not going to be very busy so you don't actually need "clearance" to push and start. Very often these airports will have drive through stands as does LIMZ.
In my case when I take-off from LIMZ in the VC10 I do push back simply because it is a large a/c for that airport. The largest they have actually had there was a 757. The 10 needs to be pushed back because of its rear engines. LIMZ does have tugs so the real life option to pushback does exist there. So what would a pilot do in real life at an airport such as LIMZ? Probably he/she would ask for pushback out of habit. But it is not required because of the drive through stand. And in fact the 737s and airbuses that use it don't pushback.
If the airport has dead end stands with or without airbridges then it is most likely that there will also be a ground/pushback frequency as well.
So PF3 is pretty much on the ball. If you are at an airfield with just a Tower frequency then having selected the pushback option in PF3 doesn't operate. So pushback at your discretion.
If you see
here contact with the Tower is after about 2 minutes of pre-flight. I pushback by myself. If any changes were to be made in PF3 for this situation the "Pushback at your discretion" would be the only option.