My push lawn mower will start when primed, then it dies unless I tip it backwards
It sounds a lot like the fuel filter or tank outlet has an obstruction which drops away when the deck is tilted. Assuming you do have an inline fuel filter replacing that can't hurt. You can drain and flush the tank while you've got the hose disconnected.

A second possibility is the governor, which often consists of a plastic flap by the flywheel. Since that is generally towards the rear of the mower, grass clippings or rodent houses will obstruct things when the mower is flat and drop onto the governor when it's tilted. Removing the shroud and taking a look is simple.

Speaking of simple, check the mixture adjustment too. It could be set too lean. I wouldn't go crazy on blind adjustments without the service manual but the screw should travel in both directions rather than being at one extreme.