I tried operating Stairs upon starting a flight with the PMDG 737, and they work normally. Your log shows something that shouldn't happen:
TIMEOUT: GSX checkAirplane STALL
This triggered a watchdog routine that is a emergency feature that is called when Simconnect stops responding for MORE than 60 seconds, which should never happen, and it triggered an intentional restart of the Couatl engine, to cover the possible chance being "stuck" was its fault, but if that restart didn't fixed it, it means Simconnect being stuck was caused by something else.
Are you using AI Traffic injection perhaps ?