> it is right that these benefits are shared

It already is! The current system mandates that universities provide a whole range of degrees to British students below cost. One of the few ways they have to do that is charging high fees to international students. If they don't want this happening they could a) increase the cap on tuition fees or b) fund the gap themselves. Taxing the income that makes this subsidy work is just going to make things work as either the system moves closer to collapse or universities become even less discriminate in accepting international students.

(Though, also, some of those degrees are a lot cheaper than others despite featuring the same tuition cap. If you go to university to study most humanities or mathematics, then you're also subsidizing the engineering, science, and especially medical degrees)