I was someplace recently where I arrived early (probably around 9 am) and the front desk agent made a remark that housekeepers start at 9 am, which gave me the impression that housekeepers were being told to do the room that would be mine first.

If the room is empty from the night before (and clean of course), the hotel *should* give it to a guest (with priority for elites) as long as there's no attempt to evade a night's charge, which for me means that you arrive at a time when people would normally be getting up, perhaps around 7 am. You can't arrive at 1 am (or 3 am) and expect to get a room without paying for the previous night. However, I suspect that the reality is that it's somewhat of a local custom, with of course airport hotels and hotels where flights come and go at odd hours expecting to be more flexible. I've seen hotels (IME the worst offenders are in Italy) pretend that all rooms suddenly become available at (and not one minute before) their official check in time, but these are places where I will never return.

In some top hotels, you can arrange your own check in and check out times. Raffles did this for their own (RIP) top tier invitation only elites (as a published benefit), SPG did it for Plat75, and now Starriott/Bonvoy has Your24 for Ambassador level only.