There aren't the jobs there used to be for students / youngsters, so I think it's quite simplistic to look at it from a Gen X point of view.
I worked from 12 (paper round), then in shops from 14. I went to uni at 18, but dropped out and went back at 20, graduating at 23. I worked mainly in the summer (factories / offices / parking and collecting new and second hand cars), but not as much in term time.
I temped in offices a bit and my graduate job started 15 months after I graduated (got dicked about with a start date).
I then worked for 4 years and then went to law school, but had to then work my way up the greasy pole before finally qualifying just before I turned 40. I had two mat leaves during my training contract.
So, even as a Gen X, I think your description of how it used to work is a teeny bit simplistic.
The problem is the lack of jobs for youngsters to do, or that people will take them on for. I got office temp jobs easily as a student, which I bet doesn't happen now.
Sending all the kids to uni (or even 50% of them) is a stupid idea. We need more vocational training from age 16. That's the way I'll encourage my kids to go. Oxbridge isn't needed, IMO. We shouldn't encourage kids into masses of debt if there are other ways to get where they want. Especially if they fancy a non-vocational degree.