Bubbles, your advice seems bonkers. Grammar schools in and around London are stupendously competitive. Both my children failed to gain 11+ places, albeit at Tiffin not St Olaves, but were offered places at "top" academic private schools. We are not talking about kids who should be considering vocational BTECs but kids who should be aiming for good RG.
What I have seen happen, and I do know of pupils who have been squeezed out like this, is that somewhere between 11 and 17 the hot house culture has not suited some bright pupils and they have started to under perform. (Though equally we have known boys who have done very well at St Olaves - it is a specific culture and some fit and some don't.) Losing a place at the end of Yr 12, because of disengagement, is a disaster. It is very very hard to find a place elsewhere, teaching a compatible syllabus, and it really does scupper chances of a successful UCAS application, limiting the life chances of otherwise bright boys.
St Olaves are very selective at 16+. I accept that in any school there will be some who don't take advantage of opportunities, but given the consequences for the pupil, I think having offered a sixth form place, the school should be very hesitant about taking away places mid way through.
Oddly I think it is different for private schools, who can turn to parents mid way through and say that a child would be wasting their parents money by staying on. The parents then have the option of spending the same amount of money on a year at a crammer. An option not available to many at state schools.