Actually, moving to the next size up of screw can be durable. Use a short, pan head sheet metal screw, the one with the big, aggressive, thread. Drill the screen door hinge to pass the threads, and the door frame can stay as-is.
Unless your door frame is really stripped out, and in that case you might want to go up two sizes if it would fit.
- pop-rivets, as mentioned above,
- Taking the door frame off (a messy job) and putting some sort of hardwood or aluminum reinforcement behind it (not recommended!) then reinstalling everything.
- Some sort of epoxy behind the aluminum frame, which can hold the screws. (also messy)
- Nuts behind the frame with epoxy (also messy)
Really, unless the hole in the frame is really big, the idea of larger sheet metal screws is sound, not likely to damage anything else, and can be permanent, since it is stronger (larger) than the smaller screws it is replacing.
Have fun with the repair.