I often hear calls for students to study near home, but don't think it is a good idea at all. For one thing, there would be a rush of people moving near to popular universities, or buying flats for their children to use as application addresses, whilst those who couldn't do this would be disadvantaged. It would be the school catchment thing all over again.
Less well-off areas often benefit hugely from having a student population move in for 9 months of the year. If this didn't happen, pubs, cafes, cinemas, retail outlets etc would lose custom, as would landlords, taxi firms and so on.
People living in areas with no universities on the doorstep would have to rent anyway, which would financially disadvantage them.
As has been said, not all universities are good for all subjects, so those living near an institution better at a subject other than the one they want to study would be disadvantaged. Not all subjects are taught at all universities, so some people would be unable to study their subject of choice without moving away.
Also, whilst students are not a homogeneous mass, and some mature later than others and have other reasons for living at home, in my experience the ones with 'helicopter' parents did less well than those who were able to make mistakes, grow up and be independent. If students all lived at home, the opportunity to find out who they are as individuals would be lost. Most student accommodation is close to (or part of) the campus, and obviously most parents don't live in these areas, so the closeness to friends and ability to attend societies and so on would be lost by those with a commute in and out.
There is far more to consider than just the rent of accommodation, although I appreciate (as a parent myself!) that this is a big expense. (NB - all of the above is speaking in response to the posts suggesting that students should study in 'local' universities, not about the current Covid situation.)