X500 will not start

I was recently gifted a John Deere X500 that had not been started this year. Service history has been great, professionally serviced. Battery was dead so it would not start.
The transmission was disengaged so the mower could be delivered, but I re-engaged it and now the mower does not move when pushed (hydrostatic system).
Oil levels are fine and I added some gas to the tank; I am fairly certain there was still some older fuel in the tank.
I attached the mower's battery to my car battery for a boost per the manual.
I engaged the parking brake and locked it in place. Fast throttle, choke all the way open, turned the key. Engine starts for a few seconds and is puffing out exhaust smoke, but then dies. After that I am not able to get much more than some initial catching and little combusted gas fumes.
What should I be investigating next?

Thanks!

http://manuals.deere.com/omview/OMM173268_19/?tM=HO

i would first check oil level on dipstick & color of oil & does dipstick oil smell like gas, if just black & no smell of gas change oil & filter, check plugs & air filter for replacement also fuel filter. drain old gas from tank & lines & replace with fresh gas & name brand fuel treatment per instructions on bottle . you may have to remove carburetor for cleaning & or overhaul kit.

https://www.greenpartstore.com/John-Deere-Fuel-Protect-Gasoline-Stabilizer-TY27534.html

https://www.greenpartstore.com/John-Deere-Home-Maintenance-Kit-Kawasaki-LG249.html

Pull the spark plugs and see if they are wet.

Spray a VERY SMALL amount of starter fluid / carb cleaner down the plug hole & crank the engine.
If it fires then do the same down the carb throat.
If it fires again then you have a fuel problem.
Check the solenoid at the bottom of the carb is working or for that matter connected.
The plug is not real good on a lot of them & I have had some fall off after I returned a mower.
If it fired on the plug hole & did not fire on the carb then you have a valve problem to be specific an exhaust valve problem.
If it fired out of the carb on either then you have an inlet valve problem.

Another quick thing to check is the safety switch under the seat. Had that happen where it was still plugged together, but pulled just enough to not let it start.

L8R,
Matt