OP, there is no such thing as “private school” in the sense that private schools differ as much as state schools (if not more so).
Nobody can usefully advise you about this particular school without knowing which one it is!
Some independent primaries are phenomenal and worth every penny. Some are money for old rope, frankly.
What attracts you to this particular school? Smaller class sizes? Pastoral care? Leavers’ destinations? Facilities?
One thing I would say (on a very general level) is that once in the independent sector at primary, kids tend to stay in this sector through secondary (unless there is some amazing state school on offer locally and / or a dearth of independent options).
Have you asked the school what their leavers’ destinations are? Is this school a prep and what senior schools are they prepping them for? Entrance exams for grammars can be quite different to entrance exams for independents. Selective independents will all have their own individual selection processes and will involve interviews as well as exams. Grammars often just have multi-choice VR / NVR tests or similar, but they don’t ask them to do creative writing tasks, group assessments or written comprehensions and interviews on top. You may find that this private school is not going to prepare them for this nearby grammar you mention as most pupils will be staying in the independent sector, so they’ll be prepping them for those exams. Every area has its own malarkey, but it’s definitely worth asking now.
As a general rule, independent preps have higher expectations in terms of behaviour. So, for instance, the head (or a senior staff member) will be in the door every morning and when they arrive they will have to shake hands, make eye contact and say “good morning.” This kind of thing. There are other diff skills that they somehow seem to absorb and, on average, they will be more confident in terms of public speaking and this type of thing. Sport will involve more tournaments with other independent schools in the area. Homework will be marked!! If you’re not happy about anything, you are a paying customer and will be listened to (hopefully - as long as you’re not mad)!
On the negative side, some private primaries are run by individuals or families who are, shall we say, ‘quirky.’ This will have ramifications for the ethos in the school. Go and meet the head and do a few visits before you decide to shift your DC. Check they are not nutters. Ask loads of questions and don’t hold back. Go with your gut feeling.
Also, your DC could almost certainly do a trial day or days before you commit to anything.
If you’re really irritated with the current school, a move probably sounds as if it’s in order. But do visit a few independents as there’s no one size fits all. Shop around basically. Good luck!