Wouldn't own a new mini unless it came with the 3 years guarantee from MINI, my mum owned a R56 Cooper S for 3 years from new and it was a very expensive car to run, we estimate about 6-7k in three years worth of repairs (all done under guarantee). She got rid just before the guarantee was up for that reason. From our experience they are a fun modern car that has "kitch" appeal but as a car they aren't that good, the coopers are better from what I can see, she had one of those without issues before the S.
Personally I would have a good look at what you want. If its classic and fun then a classic mini is a good idea, but if its reliability and no requirement for maintenance then a classic car isn't for you.
But of the two I would prefer a classic over a modern as any issues can be fixed easier and often for less. (Also I would argue the £1:£20 ratio is the reverse as modern mini parts are more expensive and labour time would be increased due to tighter working conditions of the modern car engine bay)
I am seriously considering a Cooper. It sounds like you can still have fun in them despite the reduced power. And the reduced power puts a lot less stress on clutches, flywheels, gearboxes, suspension etc etc. They are also softer spring and I think the kids would prefer the ride of a Cooper.
At the end of the day, there is no such thing as a cheap fun, quick car. I suppose that's why bikes are brilliant value - they cost very little to run apart from tyres, and speeding fines, and providing you don't bin them of course!