Any question answered...

I've never used one but they list leather as a material they cut, presumably thinish. Your local Hobbycraft probably has them and might be able to do a demo.

Laser cutting leather will make your home / workshop smell like an abbattoir, so a CNC cutter has some advantages.
However, a laser can also engrave, all the way from barely changing the colour down to removing a significant thickness ready for flooding with paint - perhaps not what you want immediately, but it could open up a few more possibilities.