The graph above shows results for the airports that participated in the 2012 UK CAA Passenger Survey and there is a very clear correlation between the UK residens share and private transport use. With such a high private transport share among local residents at MAN, I suspect it must be an attitude problem. The share number of cars may hinder any further development of the airport (no more free space for parking). Or is it an attitude problem? Why does all airports outside the South East region in this survey have a private transport share of 80% or more? Could it be that the public transport system at MAN (and the five other airports) is that bad and isn't an option to the car?
Could be attributed to several things including the facts that beyond the Greater London area all UK public transport bus services are deregulated resulting a costly, disjointed and irregular offerings.
With regards to rail in particular the majority of users are likely regional with limited to no local patronage.
This has also been observed around Heathrow.
Manchester also as you have stated has a very high proportion of annual leisure oriented flyers.
These are actually likely to prefer a private hire taxi to use of public transport even where available.
For a typical family this can be cost effective and convenient.
Then there is a simple issue of inertia prevalent with car users . Its outside the door and requires little in the way of forethought or planning.