Is there any reason I can't just remove the thermostat and upper hose, then put a running garden hose in the upper hose, and start the car for 10 minutes?

Won't all the existing coolant just come out the hole where the thermostat was?

Then just pour a few gallons of distilled water in the hose, then make sure I get 4.75 quarts of anti freeze in it!

 

You are going to make one hell of a mess...
The thermostat outlet is where the coolant will come out when the engine is running. Sticking a garden hose on it will not help - you'll be fighting against the water pump. If you can, you'll be better off rigging something to connect water hose to the bottom radiator hose going to the engine.