Of course you could, in real life you often fly more than one sector. After parking a short turnaround and fly the next sector. I have even flown routes were it literally was engines off, passengers in or out, door close, engine start. Anyway there is no argument to have that thing pop-up.
Regarding realism, I have never seen my logbook pop-up automatically after flight 
. I don’t even know where it is, I update it once every few years when I really need it for something 
.