In most distributions I've used, Minecraft just works. In Arch Linux which is the first time I saw this problem, the solution was to install gnome-keyring. Here, in Fedora 35 (with MATE if it matters, which it shouldn't) gnome-keyring is already installed. And it was already installed when I installed my system. What isn't working?

I had this issue with the Flatpak version of Minecraft, as well as the version provided on their website "for other distributions".

Minecraft used to require a Java JVM - and nothing else.

Have Microsoft stopped providing that OS-independent Java version?

It still doesn't require anything else. The gnome-keyring "requirement" is something they recommend if you're on Arch, and it's only needed so that Minecraft can remember your password and you don't have to enter it every single time. I don't know what having gnome-keyring does, and why it doesn't do it in Fedora.

Then as per the page you linked:

Well that wouldn't do anything since technically Mojang is only interested in supporting DPKG-based distributions. (and Arch Linux I guess?)

Says who? The website offers three download options, "Debian and Debian-based", "Arch and Arch-based", and "Other Distributions" - this only implies that the first two categories somehow need special treatment, not that they are the only ones supported. Though again, it is (used to be) just a Java application - all you should need is the JAR file, which you execute via an installed JVM, and that's it.

Remembering your password used to be part of what the launcher JAR did on its own without any further software needed. If that has changed, it's something you need to take up with Mojang/Microsoft.

You've paid for this software, why do you not want to request a bug fix from the developers who are in a position to do so?

Cause it's Linux! They're not gonna do anything about it.

First of all, the page you linked clearly states to not only install gnome-keyring, but also start the daemon. Did you?

If that doesn't help...
The word "keyring" is not mentioned on this page.
Isn't minecraft able to remember passwords by itself? I have difficulty understanding how java would interact with gnome-keyring. Can't you work around it with a password manager with autotype functionality, e.g. keepassxc?
Can you modify the Minecraft launcher to include the password?

I know for XFCE (on Debian / *buntu) you need to enable the Gnome services on login as well as the gnome-keyring specific things to the autostart. If you are using auto login I also needed to add the seahorse package so I could change my keyring's password to empty. The keyring can then unlock itself and login works fine.

For the record I use the "other distributions" one on my debian / *buntu. Then I can control where it installs and such. It's nothing more than a script. It installs it's own jre into the user folder I believe. I don't know about flatpak though.

^ Yes, flatpak containerisation might complicate things here.