I wouldn't be surprised if the designer of that CNC machine borrowed it from the popular Prusa MK2 3D printer, which uses a very similar design for its Y-axis belt: https://help.prusa3d.com/guide/2-y-axis-assembly_79992#81120
https://help.prusa3d.com/guide/5-corexy-assembly_835522#8648...
A few other things that can help:
- for the “it can work itself out” problem, I’ve had pretty good luck on other projects by making a standardized “dual dovetail” (bowtie) male/female slot with clearance tuned so that the male piece fits tight into the slot. Printing the male part flat on the bed means that the filament is oriented correctly for strength
- integrated zip tie slots. One design I worked on that was similar to what OP did but needed better grip… I made the slot where the folded over timing belt meshes longer, removed the printed centre pin, and put two integrated zip-tie slots: one near where the centre pin was, so I can use the zip-tie ratchet to actually pull the belt backwards and put tension on it; the other was near the mouth of the jaws, so that after tensioning it I could use a second zip-tie to “lock” the jaws onto the belt
- a third one: you make a little block with dovetails that looks like the “circle plus mouth” piece from this post, but also include a pocket for a nylok that you can use to add tension to the belt after getting the belt secured inside the clamp.
I have tried printing teeth that would mesh with the belt but didn’t get particularly good results from that. Possible that newer printers have better resolution and could print the teeth better, but my test with that just ended up with a little bit of a rougher surface and the teeth couldn’t actually mesh.
Conversely, with this design the center pin could break loose and the belt would still hold, because the pin just has to stop the loop from fitting through the slot.
Belts will burn through polycarb if it isn't tracking properly. They are strong enough to deform any pulley mount on a tight fit.
Only minor complaint is that the teeth will slip if you don't have a ton of them engaged (or they are excessively loose) but otherwise they are amazingly useful.
I’ve recently been working on a mechatronics project that involves constructing an X-Y cartesian motion system. I opted to use stepper motors to drive each axis, and timing belts and pulleys for translating the rotational torque of the steppers into linear motion. The timing belts need to be anchored to the component that is being actuated, which in this case is the carriage on the linear rails. Though matching toothed clamps exist, it is common, particularly in DIY projects, for timing belts to be fastened using zip ties or custom clips. These solutions leverage the fact that the timing belt, when looped back on itself and pressed together, creates a locking mechanism. As long as you have a tie or clip that can hold the teeth in their interlocking arrangement, the belt will remain fastened.

These are fine solutions, but they do require additional parts, and they can be prone to slipping if not attached tightly. An alternative option, which I first encountered in this DIY CNC drawing machine build (raft mount STL file here), is to design fasteners into the actuated component itself. As seen above, the fastener is effectively just a circular recess with a spindle for the timing belt to loop around, and an entry shaft that is approximately the width of the belt when its teeth are interlocked. I somewhat surprisingly have not seen prevalent use of this design, and I wasn’t even sure what to call it, though I eventually settled on “integrated loopback fastener” (let me know if there is an existing term for it).

Attaching the timing belt while maintaining tension can be somewhat challenging, but it is made easier with a pair of needle nose pliers. Once in place, the belt remains securely attached, though I imagine any sort of twisting force could lead to it working out of the recess over time. Outside of the elimination of additional small parts, the ease with which the belt can be detached and reattached is also a nice attribute. While iterating on this project, I frequently disengaged the belt by detaching from one of the fasteners so that I could adjust the pulleys and other components.

Though there are many options available, if you have control over design of the actuated component, the integrated timing belt loopback fastener may be worth considering.