Maybe avoid the "elitism route" by also implementing a proportional cap on fees?
You seem worried about elitism as you deem it, and frankly I am not sure you can ever stop it, and likewise I am not sure in your scenario it even makes sense to worry about it and instead leave the market to take care of it itself -- rich people tend not to have very large families enough to go full nepotism (though it is very much a thing) and that will leave companies crying out for trained people and thus someone paring things back to give them an education (basic office building, 10 lecturers, 10 assistants, 5 admins) and 3 years x 300 students paying 10k a piece so 9 million before taxes just from that alone... I can do something with that in almost any field and, workshops aside, scales pretty well if going lower. Paid to do research and a few inevitable tax breaks/funds on top of that...
Back onto elitism, and also where the various Nordic models will come short.
I alluded to some stuff I did earlier but go with an example.
I mostly do work for a handful of businessmen (build things, fix things, deal with standards when they want to import/export). One of his kids just got let out of a decent university with what would on paper be similar to what I have/do. He gave him a task or three within that and when the... difference in output quality became apparent I got a call. I did some stuff, and also called up the machine shop (who realised who it was, as in "you know the client 10 of big jobs this year I was point on?") and they suddenly had a free apprentice, a favour from me, and also proceeded to "give him an education" (no school like the old school and all that).
6 months later we are done and he toddles off to an interview. The classic old when the hills were young guy they keep around, and sort of person that if I am going head to head with I am up all night for weeks making sure there is absolutely no avenue of attack or weakness on my part, he actually managed to handle by virtue of the efforts of myself and the machine shop where before... no hope of that one.
Potential is still considered (the 19 year old said machine shop had found around a similar timeframe and had the potential realised, and realised in the other meaning of the term in the years since) but for my client's that sort of thing would not have happened anywhere near as quickly or decisively as what happened had the business type not opened up the wallet to spot that one (cost was equally not all that much to him -- think "semi fancy car for graduation" or "here is your house downpayment" and you might even be too high depending upon where that 20% down is). You are never going to stop that one, never going to stop the "can take an unpaid internship" set, daddy's and daddy's friends that you have personal numbers for will drive some business/contacts our way... type deals and frankly I am not sure you would want to.
At the same time I also had to query the nature of the education (I also collect old books, most of what we taught, and said kid came back a couple of years later (having seen several more rounds of recruits, even getting volunteered to shepherd a few) to say "I get it now" (old school methods are not noted as being easy on the would be recipient), was fairly common and arguably would have been known anyway as back when you tended not to be lectured to before you were time served in the lesser fields) but that is a different debate*.
*or maybe not. Law school graduation vs bar pass rate, architecture school vs architecture reality, medic school vs medic reality...
Plenty of businesses will also be ignoring degrees from various schools as it stands today both "elite" and "we don't care about community college", and plenty more will have "can you do the job?" as a bigger concern.