Yes, you will need a tutor or DIY - doesn't matter which if you are capable, can't rely on school alone, even if it's a private one. I personally prefer DIY simply because I know exactly what needs to be done for my child and I am actually better than a tutor and much more obsessive about each school's requirements, exam formats, etc
BUT I don't have the time so I do Maths/Reasoning myself but hire one for English. BTW, it took me 3 agencies and 2 "so so" tutors before I found my current, absolutely brilliant one. So don't be shy of getting rid of tutors who don't do a good job.
Private schools do have small classes and different sets, but from what I've seen top sets go faster, not deeper so no use for trickier parts of entrance exams. St Paul's papers don't ask you to have level 6 in maths, but want you to be excellent at problem solving - even at top sets it's a very small part. May be there are private schools who do extension on a regular basis and do lots of maths/maths puzzles of "Primary Maths Challenge", St.Paul's, trickier ones from North London consortium, Dulwich etc. Ours doesn't. Hence, you or tutor has to do it.
Reasoning is one lesson per week and not always in he right format - so you have to use a tutor or DIY. Of course, states schools don't do it at all, but 1 hour a week is nothing!
Private has nice touches like sports and games several times a week, playing matches, 10+ different orchestras, inter-house competition, debates, Latin from Yr 5/6, strict behaviour policies, emphasis on manners, eating together with older pupils and teachers at lunches will proper cutlery and cups for table manners, etc. Is it worth £15,000? Yes, if you have it, if you have to make huge sacrifices, I wouldn't! Also depends on whether the primary you are likely to get is OK. My friend's kid came back from a highly coveted school in Wimbledon the other day (Yr4) with a very enriched vocabulary - the word faggot was the mildest on the list. He was asking his mother what all those words meant as he overheard two boys calling each other that (and the rest) during a fight. Don't have to deal with it at our school! Is it worth £15k?