Yes MINI's have hydraulic valve adjusters so cannot be adjusted manually. You may have been told this by an idiot service advisor - a common problem is that they don't know anything about cars so cannot describe the work carried out.
BMW's protocol for diagnosing oil leaks is to clean off the engine first, then run it for a few minutes to be able to see exactly where the oil is coming from. BMW/MINI use 0w 30 Castrol Edge oil and it is just too thin for the MINI engines (which come from the likes of peugeot). The common oil leaks from MINI engines (not the BMW diesel engine fitted to newer MINIs) are: Sump gasket, diff output shafts, timing chain tensioner. It sounds like you've had a more uncommon oil leak, but the protocol should still have been carried out to correctly diagnose the leak. Sounds like you ran into some incompetent 'technicians'.