It is absolutely possible to get into a top selective independent senior school from a state primary, but only if you understand the process.
Common Entrance (the test that was historically used by independent schools for selection) is now considered a bit out of touch with reality. A lot of independent schools still use common entrance but see it as less important than before, while many have stopped using it entirely for admissions and only use it for setting. And this includes some academically selective schools. I am not suggesting that the academically selective independent schools you are looking at will not require common entrance, but I can almost guarantee that it will not be the only thing they look at.
Much more important to many schools now is the online ISEB common pre-tests, which pupils take in Year 6 or Year 7 depending on the school they are applying to. These are supposedly impossible to prepare for, but I know that DS's prep has specific time allocated in the timetable to help pupils get ready for pre-tests, and I doubt they would do that if they did not see a benefit. I would suggest that as a minimum having sat practice tests must help to get a very stressed 10 or 11 year old understand what they are doing and what is expected of them when in a timed test. The questions tend to be multiple choice verbal and non-verbal reasoning rather than writing lengthy answers in a traditional exam, but I am sure there must be common styles of question that get easier to understand with practice.
Independent schools often place a lot of importance on the interview, which is something anybody can do well at if they are prepared. Again, my DS's prep school has specific sessions on interview technique and all children have practice interviews with the prep school headmaster. Obviously they will still be nervous when they enter a very grand office to speak with a very nice but important looking person, but having experienced this and knowing how to answer key questions must help. Any good independent school will make the interview as stress-free as possible I am sure, but still......
The final key stage of obtaining a place at senior school is the reference from the prep headmaster. While any good headmaster can write an excellent reference, I wonder if an overworked and underpaid head in a state primary will take the same amount of time and care on this reference that she or he doesn't have to write for anyone else, or have the same experience in terms of what does and doesn't work, as the head of an independent prep who knows that these references are not an extra - they are a core part of her or his job.
So all that said, I have no doubt that a little genius can get into a selective independent senior school, but I think going to an independent prep makes it that much easier.
A good prep is in constant contact with senior schools and starts engaging with parents and DCs in Year 4 to prepare them for admissions. DS is in year 5 and last year we were given a pretty clear idea which schools the head thought would be a slam dunk, which he could have a good stab at, and which would be perhaps a reach too far, and what we need to do between now and Year 7.
Good luck whatever you decide!