Derby I haven't lived in but know people who have. It's an old manufacturing town and hasn't much to offer graduates. Or anyone really. The Peak District is great, and if you can live out that way and commute in then do it. But again, you won't have similar people for local friends.
It's also one of the wealthiest areas outside of London. But house prices in the really nice parts of Solihull are also high.
Saying the peak district is good for young people is like saying there's a great lake near Detroit, it's not exactly what they're after.
When I live in London I didn't drive, which was kinda nice but also meant I've only been out of city like once a year.
Sitting in traffic sucks of course, but driving rurally opens so much.
As for weekends - driving and hiking I guess?
For example I lived not far from Putney. Putney to Windsor & Eton Riverside takes 39 minutes and costs £6.90.
https://www.openstreetmap.org/#map=11/51.5330/-0.1146&layers...
Is it as glitzy as London. No. But saying there is "no culture" is just absolutely asinine.