Every other person thinks of this... and it doesn’t work. (apart from very very light surface milling)
Why
CNC machines use lead screws, as it requires power to force the cutter threw the material. belts will stretch and move or worse, and the motors aren’t that strong.
Printers generally don’t use lead screws as they are expensive and slow.
The next thought people have is lasers, but you need a high powered laser not a toy that you can mount on the carriage. (unless perspex is very thin)
People also don't think about the smoke and toxins lasers produce, let alone the safety concerns.
NB toxins from lasered acrylic can be very harmful.