What's the cost of this? And who pays it (e.g. each individual student, the school/department/grader, etc)?

I like GitHub a lot, but I will say they can be rather pricey. $7/month for their smallest private plan, you can literally run your own VM in the cloud (e.g. on AWS/Azure) for that amount (and run Gitlab), get a terabyte of cloud storage (which you can store hundreds of Git repositories in), or go to one of their competitors.

And, yes, this absolutely does require a private repository if you want to avoid cheating.