Probably too far from where you are, but there's Claremont in Esher which is non selective but has a good reputation locally as a friendly school where pupils thrive. And it has lovely grounds.
City of London Freemans' at Ashtead is lovely too. Co-ed, great for non-academic subjects -really supportive of art & drama pupils. It has wonderful facilities. (We looked at it and really liked it, but it was too far away from us, geographically.)
Reeds is getting more academic. I know several boys who I'd expect to walk the exam who didn't get in this year, and weren't even placed on the reserve list.
There's St Johns in Leatherhead? Don't know much about it, but worth a look. Less academic than Reeds and RGS.
These schools all have entrance exams, but some are more to place pupils' ability rather than to turn them down.
OP you are right that for 11+ you can apply right up to the deadline before the exam - usually November before the exam. 13+ seems far more complicated.
RGS is academic. 30% of their 6th form go onto Oxbridge and most of the rest to London, Durham or other Russell Group unis. But if your son is expected to get levels 5-6 in Yr 6 sats, then he should be fine.