I think it should be a privilege.
If higher education becomes a right, then a graduate diploma becomes something commonplace and widespread, reducing the worth of the diplomas themselves.
This phenomenon is what leaves lots of graduate students without jobs, because the employers simply no longer trust the education they receive, and there's also too many students graduating.
In some sense, diplomas are becoming more like high-school certificates, something not everyone possessed back then but something that is growing too commonplace to be of any significant value.
My views may be grimdark, but I hope you can find some truth in them.