Not certain that "private pool time" means a smaller pool for the exclusive use of your family?
We stayed a few nights at the Ritz Carlton Abama (Tenerife) a few year ago. That was 21-24 November, and it was often cool and windy, but some hot times. This ticks most of your boxes as stated. But your criteria may overlook that the two beaches that serve the Ritz Carlton (small but nice beaches) are nominally ... gasp! ... public. (I forget, could one be sand and one pebble?) And Tenerife is a massive island, with long distances between hotels and airport, and between attractions. Huge parking lots at the remote Ritz Carlton. The island also has a lot of, er, wilderness, I mean steep slope hiking trails with some danger, that kind of thing. The island also has All-Inclusives that attract lager-louts and other undesirables for your criteria. Not at the Ritz Carlton, of course.
As a big fan of the Caribbean, I'll note that we often met small numbers of Europeans at our high-end, "Old Caribbean" resorts. Including, as I pointed out to my wife, German grandmothers in bikinis. But, "Old Caribbean" means no children under 15 in high season. Alas, that rule is fading, due to income/business maintenance pressure, especially from the Masters of the Universe.