When I switch FR24 on it will open at the last location I had it at when switching off. That has always been the standard with me since I've been using it.
A while back they started masking many private flights or switching off tracking when under a certain height to obsfucate the departure/arrival point.
Many private (executive) flights are "masked" at the request of the operator for reasons of privacy and/or security. The disappearance of labels close to airports can sometimes be due to the coverage or settings by the provider. Remember that FR24 relies on subscribers feeding them with their SBS data. Those subscribers can differ in their own personal settings and this will be reflected in what you see.