may be as simple as a stopped up gas cap air vent. Try loosening the fuel tank cap and trying again. If Ok, use a thin stiff wire and compressed air to probe the vent right in the center of the cap.

This happens to me alot when the fuel in the tank is low. I have to turn the tractor around and face it in the opposite direction so the fuel pump can get some gas.

Sounds like you have the same problem.

Thanks for the advice! Based on your advice, I tried again this morning. Now the engine is starting up fine, but appears now to be cutting out when I dis-engage the parking brake. Engine runs fine until point when I release the brake, in multiple test runs.

Am I looking at an electrical issue? Does what I'm seeing now make sense in light of my engine dying on the hill yesterday? Confused

That "typically" sounds like a seat switch issue.
Without knowing which model and not having a schematic, I don't know if your SS is NC or NO.

Make sure it's fully plugged in & plunger travel is properly adjusted.
Maybe with the slope, you are leaning a bit too far in one direction?