Great question, and I hope other will chime in with their thoughts on this, too.
Here in the US, we have a good many 'snobbish' organists who put on such an air about themselves as being so very special - the rare gift to their trade - and talk in such a manner replying to a question with "we organists feel ... blah blah blah".
On one occasion or another, while trying to have a conversation with these snobbish types, look right through me like I'm a piece of glass, trying to catch glimpses of whomever may be noticing "them" ... how rude, indeed.
The situation of not being able to play these snobbish types instruments is true, even here in the US. In some cases, unless you have a doctorate degree in organ playing, you are nothing, and are not 'worthy enough' to play, let alone look at the console.
Sorry, but I have a problem with that ... the fancy scrolled paper with someone's signature that nobody can decipher does not make the musician ... the musician makes themselves a musician.
Unfortunately, those situations stereotype the profession in some ways as people always tend to dwell on the faulty points of our profession rather than all the good we actually do.
I do not have those degrees in organ, but I have been playing (and paid for it) in the Lutheran church for over 50 years, as well has having the good fortune to have played a good number of prominent organs here in the US as well as Europe.
Point is, I am neither better nor worse than those organists who have studied organ at the university level.
I am always elated to show anyone, who comes up to the console after services, anything they want to know about the organ - and even offer them the opportunity to play it.
Ours is a dying profession (church organist) and few are coming up through the ranks to take our place when we finally have our organ shoes bronzed and mounted on the wall. I began playing in church at age 12 ... few, if any, 12 year olds in this day and age would make that same kind of committment. What a pity.
I've never regretted that first opportunity to play in church ... and it happened only because the regular organist got sick that particular Sunday.
I don't have an answer on how we can remedy this tainted image