Two good mates of mine have Minis and I have looked into one as a first car so I do have an idea of what I'm talking about. One of my mates found that he couldn't get anybody to insure him on a Mini registered before the cut off for standard insurance companies (normally some date in the '80s), this restricts you to newer cars which are typically in better shape and more desirable than the
rougher older examples. Insuring a Mini is not that cheap either, I know my mate was paying £2000 for an 850 at 17 and was paying the same again when he added a few (non-performance enhancing) modifications to it at 19. My other mate bought a '70s example for £50 has spent over £1000 on parts to do a shell up restoration on it and having realised the issue with insurance he obtained a log book from a newer car so he could get round it (even though he'll now have to pay tax on it) in the end he has decided to put the project on hold because he doesn't want to build a standard 850 and anything else would be too expensive to insure so he has bought an old banger for now.
Insuring a Mini isn't that expensive but you'll need to be looking for a late 850 that is completely standard (relatively few are) and purchase price isn't that cheap either. If someone wants a Mini they'll get a Mini but as a cheap first car a Mini isn't a terribly good option. I wouldn't bother trying to find insurance for a classic at 17, I had no luck trying to find someone who would insure a standard 1959 Morris Minor 1000, a car that is far less likely end up in a ditch than a Mini or most small hatch backs...