TiggyR The Milk Round does indeed exist (or did when I was graduating, in 2003) and those of us with 'useless' degrees in English Lit, French, Philosophy etc were being wooed, wooed, wooed by fancy firms in finance and PR and whatnot.
As I understood it then (and I work with undergraduates now, although in America, but I think it's still true) what matters is: Getting a good degree (2:1 or above), doing something else at Uni other than socialise and work (plays, Uni newspaper, charity fundraising, Uni student government) that shows initiative and drive and ambition, and then using career etc facilities at the uni, and working as hard as possible to get summer work experience, internships, and so on.
Frankly, most big employers said they didn't care what your degree was in, they wanted it to be from a 'good' uni, at a good grade, and not the only thing you've ever done. The message seemed to be that getting a BA was considered evidence of commitment and a worthy achievment in and of itself.
Certainly, I don't know anyone who did an Economics or Business Studies degree who thereby got a massive legup in the business world. Conversely, I think the other stuff matters a great deal more than folk realise.
As for the OP: I'm torn. I was terribly independent at 18 and would have died inside had my parents insisted/offered on joining me around open days. I now work with undergraduates who have no shame at all about apologizing for being late by saying "My mother usually wakes me up and I just can't seem to manage it myself."
(I confess, I icily said "Well I suggest you get a better alarm clock, or move home") - I think that over here in the US, where parents are forking out around $40,000 per year for the 'best' unis, the students are much more coddled. At least in the UK by the time they move out of halls, they've worked out how to tie their shoelaces. Over here they live in all the way through - and so have to learn how to get a flat, etc, when they're jobhunting!