Exactly the same point can be made about private senior schools. There are plenty of excellent state comps in the area and DCs there will do as well and even better than pupils at private senior schools.
This is very true. DD had some specific wants that made a private school more practical but they could also have been achieved with outside help at a comprehensive.
could help her more with home work
I've never helped her with homework! She's hardly ever had any, bar reading and times tables.
Re exam technique, it is honestly kind of common sense. But ten year olds are not really known for that particular quality so you might as well explicitly tell them!
Firstly, pace yourself. Have a quick read through the paper, identify (if possible) the questions that have lots of marks attached so you know where to spend the time. Make sure you know if there is say, a long question in the middle of the paper followed by lots of short ones or a long one in the middle and another at the end. You need to make sure you have enough time to do those long questions which usually have more marks attached. If there are two parts to the exam (like DD's SPGS English which was a comprehension and then a longish creative writing task), decide how you are going to split your time.
Leave some time to check your work at the end, too.
Second, if you get stuck on a question, don't spend ages getting bogged down in it. Move on to one you can do. You can come back to the one you couldn't do at the end, or later on if you have a sudden inspiration which does quite often happen.
Read the questions really thoroughly and more than once. Make sure you know what they are actually asking you to do. Answer the actual question, not the question you would prefer to be answering (more applicable to English etc but also possible to end up calculating the wrong thing in Maths).
Show every bit of working or every step of your logic. If you are not sure how to set it out, write a short sentence or two explaining what you are doing. Even if you get the final answer wrong, you will get some marks for knowing how to do it.
If you are running out of time, condense answers to the basics - eg use bullet points if you can. If you are really really running out of time, prioritise questions you know you will be good at if possible.
That's more or less it, I think.
Oh, another thing. Wear layers if you can. Being too hot or cold is really distracting so make sure you can cool down and warm up if necessary. Obviously not so easy if school uniform has to be worn. I did not send DD in uniform to her exams but I can see some schools might like you to.