Electronic – Best tool bits for PCB cut outs (board shapes and slots) when prototyping with desktop CNC milling machine

Why don't you look at www.linuxcnc.com. The students could also learn to wire some actuators and sensors, with 12k$ budged you can build good quality router with ballscrews and precise linear guides. Perhaps the PCB milling needs somehow lighter spindle for fast movements compared to metal working, that would also require liquid coolant.

What would be the root cause of such issues

A router will produce dust as it operates. Some of the dust will end up on your board.

There might also be a sanding operation to smooth the board edges after depanelization, where dust could be produced.

The dust should be fiberglass (assuming you designed the board with clearance of copper from the board edge), so it shouldn't normally affect the operation of your circuit. If you will use your board in a clean-room environment or something you may have other reasons not to want dust on it.

and how can we prevent it from happening again.

You could just talk to your vendor --- maybe they just need to adjust the dust collection system at the router machine. But this will probably not 100% eliminate the dust on the board.

You could use shears to depanelize the boards rather than a router.

You could require your vendor to clean the boards after depanelizing.

You could aqueous wash your boards after assembling the parts onto them, which you might do anyway to remove solder flux.