However, these cores are designed to spin very fast, not at grid speed, so these need some form of gearbox or electronic frequency conversion.
You're right that the core doesn't spin at synchronous speed but the LP shaft does. It's optimized for 3600rpm, but could run at other speeds... the machine just isn't designed for it. The LM6000 only uses a gearbox for 50Hz units while 60Hz don't need it.
That's true of any gas turbine engine. About the only engines you can use to drive a genny directly are diesels at 1500rpm for 4-pole machines or petrols at 3000rpm for 2-pole machines (or if you're in one of the tiny handful of countries in the world that uses 60Hz, 1800rpm or 3600rpm).
You can of course run these on alternative fuels like waste veg oil for diesels or propane instead of petrol, and the latter used to be quite common on remote farms where you'd have a big propane tank for heating and cooking - may as well run the generator on it too, and not have to worry about exhaust fumes.
I don't believe that's right. Near as I can tell, the large turbines built for power generation, like the GE 9HA spin at 3000 or 3600 RPM. That's how they get efficiencies north of 60%.