I'd choose Old Palace out the 3.
As you say not a huge difference in academics of the 3 schools, but I do know girls at Old Palace share some sport and gym facilities with Whitgift and Trinity schools, also do productions together and other activities. Having 'brother' schools which have amazing facilities and reputation would be a big influence on my decision.
Also Whitgift Foundation ( that own Old Palace, Whitgift and Trinity)
give generous scholarships and bursaries, useful to know if your family has sudden change in circumstances, god forbid.
GDST schools have a super solid rep, and both schools edge out Old Palace academically. Friends tell me since Palace took over a non-selective girls school in South Croydon, Croham Hurst indepedent girls school, it lowered the academic standing of the school somewhat.
I personally think given its excellent sport
and links to elite boys schools Whitgift and Trinity, it still edges Old Palace ahead, but academically I think Sydenham edges ahead.
I think you shouldn't even see these schools as much different from Jags and the 'super selectives' - the percentages of girls getting A* to B in most London girls Indie schools are very high. We often check girl schools on our radar for 11+ choice, looking at GCSE and A level results and the differences between even say St Pauls and Streatham aren't huge at all. Looking at A level results for 2022,
St Pauls Girls A* -B : 98.8 %
Streatham A* - B : 83.6 %
That's hardly a mountain, and excellent for Streatham when one considers St Pauls is a gold standard.