I tried it with Firefox, and it remembered my login between sessions.

I don't use Chrome myself. I use Firefox and occasionally Brave, which is using the chromium engine and I don't have the issue you have. But, as many people here do use Chrome and don't have the issue you describe either, it must be something related to your configuration.

Maybe you need to delete both your cookies and cache. As far as I know it is possible to delete the cache for a specific site, but I would need to check how that is done in Chrome. However you can load the site without using the cache by having the forum page open and then using Shift+F5 or Shift+Ctrl+r, both do the same and reload the page without depending on cached information. It will take a bit longer to load the page and you would need to do that after deleting the cookies.
Besides that or an extension you have that does something, I can't tell what is the cause of it.

If you feel adventurous, you can also load the page in developer mode (F12) and look at the output that is being generated to see if there are any errors, when the page is loaded. Maybe they give an indication.

You could try opening Chrome's devtools with F12, I don't have much knowledge to debug general issues from there but maybe you do, or you'll see something.

The 'Applications' tab shows, among other things, the cookies (after selecting "cookies" on the left, you may want to type "rpgwatch" in the filter if you have cookies from secondary sites).

- I did a quick test, taking a screenshot of the sorted cookies when the session was fine, then messing with one of them, reloading and comparing the cookies.

It didn't tell much, but I saw the missing cookies which may be an indication. There's a "only show cookies with an issue" at the top, but that didn't show me anything. Perhaps you'll have more indications from all that.

- When I checked the open issues (see top right), I had two: one about quirks mode because of the header, and another irrelevant one about audit usage.

If you have an issue at this point, it's likely due to an add-on or to the settings like some restrictions on the cookies ("Sites that never use cookies") or another site setting.