Um... Lifting... that is one of the most clear and concise explanations of the free-turbine engine that I've ever read. Nice job!
One slight expansion though...
In a turbofan, some of that thrust is used directly and the rest drives the fan...
Yes, but... As you explained, in a turbofan, after (or "downstream") of the gas generator section is another set of turbine wheels. These are connected by a shaft (a separate shaft from the one that connects the gas producer and compressor) from the back of the engine to the front of the engine where the big fan is on modern jetliners. It's this
shaft-within-a-shaft business that makes the whole shebang work. It also allows for the turbines to instead drive either a propellor or a driveshaft to the transmission as you aptly described.