To pay for uni education you could raise taxes. But that would mean taxing high earners who went to uni tand high earners who "worked thei way up" and never went to uni. Which always seems to generate backlash/resentment and is therefore politically unpopular.
One solution could be to make uni education free but impose a graduate tax on future graduates such that they have to pay a certain percentage of their income once they are earning above a certain threshold. This would make it affordable to keep education free for the graduates after them. If they never earn enough to qualify for the tax they don't need to pay it. And once they reach a certain age or have paid as much as their education cost you could cancel the tax.
That would be a reasonably fair way to do it. But it's also how the student loan works in practice anyway.
Personally I would rather have funded education but that would either mean raising taxes overall to pay for it which would be too unpopular. Or would mean cutting spending elsewhere. And I think other areas (the NHS, carers, disability benefits, child benefits) are more deserving when it comes to making a choice. I do think progressively writing of someone's loan depending on how long they spend in the NHS/public service could be a way forward.