You might want to consider the FS Costa Palmas on the east cape. I haven’t been but Besvisor went and was a huge fan. It is quite a distance from the two towns of Los Cabos, but if you don’t mind being marooned, it could be great.
Kat is correct. Chileno does have a small private beach. The design of Chileno and the Montage are quite similar. The Montage beach might be a hair nicer, but the F and B prices there are astronomical.