Timing belt drives are used on some commercial CNC routers (e.g., Romaxx) and DIY CNC (
https://www.inventables.com/technologies/desktop-cnc-mill-kits-shapeoko). They can be quite precise, if not overly stressed, and are considerably cheaper than a ground ballscrew. They are not too good for heavy milling, but even on Bridgeport conversions, the linkage between the stepper and ballscrew (leadscrew) is often a timing belt, because there is a ratio reduction and the belt is not stressed. From the little information provided so far, a belt may suffice. I would be more concerned about slop and jiggle from the track and rollers shown in the YouTube example.
Duda123 needs to put some numbers to her/his design. Otherwise, we will just be shooting in the dark. This project started out being just a servo running a small trolley with tires and depended on the electronic center (i.e., the dead band). It has now seemingly evolved into needing micrometer precision.
If one goes to Servo City (
https://www.servocity.com/), you can find the dead band. For most analog servos (i.e., the ones likely to be converted to continuous rotation), you will see that the dead band is about 5 to 8 uS. The full range is +/- 500 uS for +/-60° or so. Using a dimensioned sketch and known gear ratios, one could calculate its positioning accuracy. 2uS is about as small a dead band as one can find in hobby servos.
BTW, I am not recommending or criticizing Servo City as a vendor. It is simply a single location with a lot of technical information for a large number of servos.
John