I've never heard of somebody who thought they were PAYE being declared as SE. That would be quite suprising.
Maybe there are marginal jobs which are unclear. I was suprised to learn the other day that you can be paid purely on piece rate (i.e. paid nothing if you do nothing) but still be PAYE and have all the rights. This feels completely perverse, and that's before you take the NMW into account (for somebody on piece rate!!).
The other way round, a SE person being regarded as PAYE, usually against their wishes, is not unusual.
S.A.S. - if you are SE rather than PAYE, you don't get sick pay but you will get extra money instead. That's the deal. So, the ultimate benefit of PAYE has to be the security of being able to sue. Given that any half decent employment lawyer is about £200/hour, wow much would you bother to sue for? 5k, 10k, 20k? IMHO 5k is not worth suing for. 20k will tip all but the big flying schools into liquidation; you can get the judgement OK but collecting it will be something else.
So, job security in this business is likely to be illusory.