I live in one of these towns, have done for 25 plus years. We moved out of London to have a better type of life for our children. We had one child when we moved here, then two more. We raised them all here and had a mixture of state and private schooling.
Both of us were working in London at the time we moved here (DH still works in the City prior to Covid) so we followed the train line out With good train link being very important. At the time, Walton on Thames had a fast train into Waterloo - non-stop. Not really the case now. But Walton is a main line. Weybridge has two lines but the direction of Virginia water, chertsey etc is looong. Lots and lots of stops.
Second big requirement For us was the house being walking distance to the train station. Station car parking in these towns is pricey. I’d concentrate your search On all three towns but waking distance to the station. Esher station is actually quite far out of the town centre.
Other areas quite close that have great schools are Thames Ditton and Hinchley wood. Hinchley wood school is particularly good - really as good as private schools (I know this from experience)
Ashley park primary, cardinal Newman in Walton are good, St Charles boromeo, manby lodge, Oatlands are good primaries In Weybridge.
Walton has the leisure centre, but you need to drive there so any of these towns nearby would be fine to look at as you’d likely drive for swimming classes and other classes
East Molesey is lovely and has two good primaries but the station is Hampton court which is not main line so service isn’t that great compared to Esher, Walton and Weybridge. West Molesey is NOT the same as east Molesey. Each of Esher, Walton and Weybridge have areas to avoid but you’d see this very easily driving around the area. A day out driving in these towns, driving from each train station outwards would be useful to see the areas you may like and are easily walkable to the station.