Why does 'Keep Me Logged In' not work on my 9th Gen iPad for a local paper website?

The "keep me logged in" option will be reliant upon a session cookie being created on your iPad (and or any other computer/device that you may use). Clearing your cached data will, by definition, clear any associated cookies.

Session cookies may have a maximum life that will, when created, be determined by the online service. Upon expiry of the cookie, or after its deletion, you will have to repeat your login. The App associated with the online web-service is effectively a browser front-end - with its own saved data and cookies.

Offhand I can think of half-a-dozen sites that I use a lot which ignore the keep me logged in box in both FF and Safari. It's usually just bad coding by the devs rather than something you're doing. I've chased a few of the ones I use the most and the stock answer is "we only test it on Chrome".**

The other things that can prevent it are privacy extensions, ad blockers or script blockers that, somehow, prevent the session cookie being saved, especially if the devs have farmed out the functionality so it looks like a third-party or cross-site cookie.

**I'm not suggesting you download Chrome. Indeed, I'd recommend you don't.

Do you see this behaviour with other websites or online services? If not, the problem is not with your iPad - but instead lies squarely with the online service and/or their associated App.