Sounds like you're looking at the right stuff. Report back please after you've checked the "freon". 
I’m having the exact same problem, I opened the glove box and removed the cabin filter, the blower motor will turn a few revolutions, and just like yours and shuts off with everything else. The a/c clutch and compressor are fine,heater control valve is fine, I’m thinking it must be something simple that is interfering with the start up of the system.I realized last night that I lost track of my original problem which why it won't continue to run longer than a few seconds. I got absorbed into figuring why the pressures were odd. Once I get that straightened out we can see what else it's doing.
How do you eat an elephant?.....one bite at a time.
Seeing that I don't have a clear understanding as to how this system operates and the sensor integration I will need to have a professional look at it.
I used to work on the old R12 systems in the 1970's. Much simpler than todays.
I'll be back.