Hi, JDT, welcome here. I'm no expert, and I didn't stay at a Holiday Inn last night, but I do have a BX, and maybe an idea or two for you.
First, are all the fuses OK?
Second, and just a guess, but there are some safety shutoff circuits on the BX, such as for the seat, that kill the engine like that. They are probably some of the most common headaches users encounter, because a bad switch, loose connector, frayed wire, etc. can keep the engine from running. Usually you hear about them completely deactivating the engine, not as you describe, but I'd begin with those circuits. Do you have an owners manual? It describes the safety circuits.
Third, maybe there is something wrong with the fuel shutoff solenoid or the circuit giving it power, which interrupts the flow of fuel to the engine when you turn off the key.
Related issue: This is a machine that is 6+ years old. How long have you owned it? Did it previously run OK for you? Any chance the wiring got boogered up somewhere in its life ( "boogered up" is a technical term real tractor experts sometimes use, so I'm using it to sound smarter than I really am

).
A final point... if the engine only runs with the starter cranking, you do know it's not good to keep it running that way, right? Just checking.
Maybe one of these ideas will give you something to work with, or maybe someone with more specific ideas will jump in.