At the risk of changing the nature of the thread, most local reasonably well-off Sanderstead parents view that the state primary schools in the areas are pretty good, and typically only put their children into private schools either from 7+ or 11+. The junior section of Croydon High seems to be full of girls who live further away where the primary options aren't so good. There is usually a surplus of private primary school places, so you can transfer in later if you felt that the state schools weren't doing enough for you.
The main disadvantage imo to Old Palace, Croydon High and Royal Russell is that they all feed into their own senior schools, and with a 3yo you can't tell what sort of senior school will suit her best, nor how the schools will change over time (eg Croydon High is now on their 3rd or so Head in 3 years I think and their numbers are dropping). The preprep/prep department of Old Palace isn't in central Croydon btw - it is in leafier South Croydon. Woldingham is a senior school only.
Although it has the same problem in terms of feeding into its own school I would include Caterham school on any list for the area.
In terms of independent co-ed primary schools, there is Warlingham Park as well Essendene Lodge & Oakhyrst in Caterham, and Oakwood (RC) and St Davids in Purley. These will at least prepare your dd for a range of senior schools
Other than the junior section of Croydon High I don't think that there are any independent primary schools in Sanderstead (which should tell you something!)
There is a new girls school opening next year in Purley. The old Lodge school is closing and there will be a Cumnor House school for Girls on that site, taking in the previous Lodge girls, but obviously hoping to add to them.
But to be honest I would look at the 3 Sanderstead primaries as well as Hamsey Green and Warlingham Village. Assuming that you are moving before your school application is in I suspect that you'll find one of these schools would suit you.