I feel you could make an equal case for the government/lenders being the customer.

I think you're probably right about students not being the product, they're more of a raw material?

I'm being slightly cheeky, but I don't think the relationship is best viewed as the student being the customer, certainly not if the degree is the product.

You could say the real customers are corporations, that students pay as a supplier.

in most cases these days, the degree doesn't pay, so you're right, it must be the product.