Start button is not doing anything

You may need a multimeter to check if the motor itself is dead. If it is the motor, the replacement may be available via Audio-Technica directly. The part is weirdly hard to find online otherwise. you can contact them via email at parts@atus.com. (If in the USA, check your regional Audio-Technica site for a parts page otherwise.) It will, however require a bit of soldering to replace as two of its wires are connected to another board directly instead of via a push connector.

You can see details as to how to do this in this guide for its predecessor: Audio Technica AT-LP60BK Motor Replacement

The internals are the exact same between the LP-60 and the 60X layout-wise. AFAIK the only major changes were in the physical appearance of the turntable. I would not recommend de-soldering while the motor is still in the unit (as in that old guide) as the entire shell is plastic. It's also just a big unit, which you already know due to owning one. It'd be annoying to work on it like that.

Does it indeed make a mechanical "click" when start is pressed?