The DL policy (also policy on UA and, I believe although am not certain) is that he who boards loses his UG. The software is designed to do this when the gate reader scans the BP as "boarded." The reason for this policy is that it can delay a flight's departure if the GA, then FA walk through the aircraft and then swim back upstream with a pax who may not be in an aisle seat to an F seat which may be a window, all of this when the aircraft is going through final prep for pushback.
While it's nice for the indiv (and I won't complain when it happens to me), it's terrible policy or practice to go onboard and that's why the pager idea is even worse. At crowded (ATC) airports such as ATL and LGA, missing pushback can mean losing a departure slot and a lengthy delay.
So, the choice is to remain at the gate and risk the OH issue or to board and be resigned to Y.
The real policy change which DL could implement and which would eliminate the reason for wanting to board is by enforcing its own carry-on policy. If pax weren't carrying on steamer trunks, there would be enough OH room and nobody would need to gate check. That eliminates this issue and frankly also eliminates the gate lice and mob scene for another.
In the day, F was boarded last because nobody wants to sit through the boarding process.