If I really wanted them to get into that school (which it sounds like you did), then yes, I would have.
Honestly though, school sounds a bit wanky. The second reason makes a bit more sense, but the first part, that their exams are so super special and unique that coaching won't help, is just nonsense. There are only so many types of questions school related tests can ask!
Even if the actual questions are slightly different to other schools, all kids will benefit from practice and tutoring in how to read questions properly, how to write clearly, how to formulate answers, show workings out, etc. Not to mention general subject related stuff like going over maths, grammar, and so on, all of which if a kid is shit-hot on it could make the difference between meeting the required score or just missing it.
Regardless of the questions they ask, a super bright kid who doesn't pace themselves so writes loads for the early easy questions and then doesn't have time to do the final hard ones, or who writes incredibly well but goes of on a tangent and doesn't answer the actual question set, or who solves the complicated maths question in seconds in their head so just writes the answer,
might miss out to a tutored kid who completes the whole test, answers the exact question and includes their working out when the question specifically says to record this, etc.
Fair or not, there's a reason most families do tutor!