My dc is also year 5 and from state. My eldest is year 7 so we have already did the 11+. She had a good experience. She is naturally bright and only started prep end of year 5 bc we did not realize until last minute that we would do the 11+ All went well for her. We started prep earlier with my ds, as we knew he would do it since his sister went through it and he wanted to. Anyway, it is tricky to figure out what is the right amount of prep, as circumstances are different for everyone. For us we have the opposite, my ds does better in 11+ math, but he is also in a bilingual school, so half his school life is taught entirely in French. Therefore, losing out on a lot of English.....so we will need to start picking up our English reading and comprehension efforts. My dd also had the same, she was better in math....but because she started prep end of year 5 her scores seemed to jump quickly.
And to answer your question for the private schools, some schools do not set much homework and parents give their dc extra homework. I know this from family member who sends their dc to a top girls private school. She said all the parents give extra hw, some sending dc to verbal reasoning workshops as young as year 1!!!! I do not think this is all necessary, but I think they want to ensure to get a spot at a top school.
As for down time, both my dc do sports 6 days a week. Dd continued this during her 11+ prep, exams etc. She did not want to drop anything. And she is the same in secondary school. But many parents drop things (from what I have seen). My dc also watch tv daily, use screens a little too much for my liking. But I did read someone mention that the dc that had a no screen rule during the week got into the grammer schools, where they ones that were allowed screens were less likely. DD did not apply to any grammers, only to Independent schools. DS will also only apply to Independent.