For the spindle motor, you need at least a basic SCR drive. Most common are KBIC drives, but you have to be aware the control circuitry is non-isolated, and floats near mains voltage. You can get isolated versions.
IIRC Sprint also do fully isolated SCR drives. If you do a search on RS for "Sprint drive" they have various options.


I've not watched the video, but I'd guess the X-axis motor is simply stalling. It's what happens if you try pushing a stepper motor too hard. Either drop max speed, or drop acceleration.