I have some experience of St Katharine's - my child attends this school.
Positives first: The buildings, facilities and the grounds itself are excellent. Plenty of space, lovely hall, swimming pool, sports field, trim trail - what more can you really ask for at primary level? A new classroom is being planned so that, in future, there will no longer be any mixed age groups in classes. Enthusiastic PTA - and the children are generally pleasant and well-behaved.
That said, I think you've hit the nail on the head with your use of the word "tired". I feel this is a school resting on its laurels. It was highly over-subscribed a few years ago, but this is no longer the case - a number of children are "allocated" this school each year without having named it as a choice on their application form. (In fairness, it has increased its annual intake from 15 to 30, so that may account in part for the change in status from "over-subscribed" to "under-subscribed".)
From what I gather from conversations with other parents, a large proportion of children have tutors from Y3/Y4 onwards to get them through their SATs and the 11+. Maybe this is pretty-much standard for a fairly "middle class" school near Sevenoaks - who knows? I just didn't expect this - I mean, I knew about 11+ tutors, but not "general" tutors.
This is an okay
school - sadly, I'm not sure that any of the alternatives to people in this part of Kent are that much better. (How do I know? I've looked into this very closely given my overall disappointment with the education my child is receiving). Of course, some - perhaps many or even most - of the parents there think it is a wonderful school - I'm just giving you my own, personal opinion.
Hopefully, some other people with experience of the school will come along to give you their opinions as well so that you will have a fulller picture to help you decide whether the school is the right place for your child. Hope this helps.