The 'train in one school' idea started with some comments by Ian Cheese from Flybe when he was talking about the introduction of their 'mentored programme'. The argument was that it was easier to keep track of the training records if they were all in one place.
Ian readily accepted that PPL, hours building and ATPL groundschool were likely to be done at different FTOs. AFAI remember he was encouraging the 'structured modular' concept where one FTO, usually the CPL/IR training provider, arranges all the training, is responsible for quality control and, crucially, maintains the training records. Note that, 'train in one school' in this context doesn't literally mean 'use one FTO for everything', it means 'have one FTO maintain your training records'. I vaguely recall some mention of this in Flybe or BA Connect recruitment literature of the time, about three or four years ago, someone may be able to remember.
The concept seems to have been latched onto by some of the larger FTOs (who can offer all stages of training) as a vague justification for attending their modular programmes. In reality, very few airlines care where you trained.