From memory, Prince Andrew's naval career finished because he had served all the years of his intended commission - in other words, it expired because of effluxion of time. Although reputedly not very academic, the belief expressed in the newspapers was that Prince Andrew was an adequate Naval Officer.
I think that the reason that members of the BRF tend not to work is more a question of custom-and-practice. I should have mentioned that the Duke of Gloucester was a successful architect and no doubt would have continued his career except for the fact that his elder brother William [who was of course in line to succeed to the title] was killed in an aircrash [a sports event IIRC]. Until that time, Prince Richard [as the Duke of Gloucester was at that time] as a 'younger royal son' was not expected to have a Royal Career - just like Prince Michael of Kent.
Sometimes, I think that lack of qualifications is a reason why some members of the BRF would not be able to hold down a career [i.e. that did not depend on their royal status] particularly in the case of female members. Diana, Princess of Wales memorably failed to achieve a single 'O' Level. This was a national exam in many subjects and I have seen her lack of success described in an American Magazine as 'Diana was a high school drop out'.
Sarah, Duchess of York achieved 6 'O' Levels and attended a well-know private secretarial college [Queen's] in Kensington, London. This college is now defunct; entry requirements were minimal - bascially ' Could your father meet the College fees?' The former Sophie Rhys-Jones acheived both O and some A levels [an exam for 18 year olds] but did not achieve sufficiently high grades to enable her to go to university, although her career path was always going to be Secretarial through choice; she then moved to PR, her career move [from Secretary to Principal] being achieved just as her friendship with Prince Edward became widely known!!
The new generation of Royals [William, Catherine, Beatrice and Eugenie] are all either university graduates or intending to be, so yes, in one sense the BRF is becoming better educated and could obtain some form of 'employment based on their qualifications rather than their position'.
Viscount Linley, although technically not a member of the Royal Family, trained as a skilled carpenter craftsman; he now runs an upmarket/upscale furniture making business, known as Linley, and headquartered in Pimlico, London but acts as Chief Executive rather than craftsman nowadays. When the Queen goes on tour, she instructs the Foreign Office to buy from Lindely, thereby helping boost her nephew's business.
Some years ago, one of the tabloids produced CVs [Resumes] based on the known acheivements of BRF members, and using their exact education, qualifications, work record and ages etc and then submitted these CVs BUT UNDER FALSE NAMES to various employment agencies. From memory, the Princess Royal and Diana were given the 'existing occupation' as 'Charity worker' to reflect the charitable roles undertaken by these ladies. The result was that no Royal managed to secure any interest whatsoever, other than Sarah, who was offered an interview for work as a barmaid, her secretarial skills being considered 'too rusty' and also unexceptional [she apparently came last of her class at typing and shorthand at College] to gain her work as a Secretary.
Alex