The funny thing is even fusion plants would produce energy by turning a steam turbine.

Not useful ones.

Predicting here there will be no fusion plants that drive steam turbines, ever. (Unless General Fusion's system turns out to work.)

How else could a fusion power plant generate power?

D-H3 and H-B fusion release their energy in a form that can be extracted electromagnetically, with no detour through heat and turbines, so at least in principle could be cheaper to use than fission, unlike the D-T fusion that ITER etc. target.

There are projects working on these. They have much higher activation energy, so are harder to make happen -- for hydrogen-boron fusion, much, much harder. Helium-3 is scarce, effectively available only from decay of tritium, all of which has to be synthesized, then it needs to decay, with a half-life of 12 years.

It's even worse: most generators transfer mechanical power to electrical energy.

Hydroelectric plants, nuclear reactors, gas and oil plants, all of them depend on mechanical movement.

Mechanical power is usually taken to imply a gear, belt, chain or other lossy linkage. Just rotating a shaft is lossless, and should not count.

What is so advanced about it? We’ve had pressurized water reactors since the 50s

Yeah, we're pretty close to the point where as much time has passed since pressurized water reactors and boiling water reactors were invented as there was between their invention and the invention of the steam turbine.