I think you are a little confused at the problem here. The issue is that partial updates are not supported. The reason for this is very simple, Arch ensures that any given package list works on its own, but not that packages from different versions of the package list work together. So if Firefox depends on libssl the new Firefox package may depend on a new libssl function. If you install that version of Firefox without updating libssl it will cause problems.

There is no way around this limitation. If you install that new Firefox without he new libssl you will have problems. No matter how you try to rules lawyer it. Now 99% of the time this works. Typically packages don’t depend on new library functions right away. But sometimes they do, and that is why as a rule this is unsupported. You are welcome to try it, but if it breaks don’t complain to the devs, they never promised it would work. But this isn’t some policy where you can find a loophole. It is a technical limitation. If you manage to find a loophole people aren’t going to say “oh, that should work, let’s fix it” it will break and you will be on your own to fix it.

Focusing on your commands. The thing is that pacman -S firefox is always fine on its own. If Firefox is already installed it will do nothing, if it isn’t it will install the version from the current package list. Both of those operations are supported. Also pacman -Rs firefox && pacman -S firefox is really no different than just pacman -S firefox (other than potentially causing problems if the package can’t be allowed to be removed due to dependencies). So your command isn’t accomplishing anything even if it did somehow magically work around the rules.

What is really the problem is pacman -Sy. This command updates the package list without actually updating any packages. This will enter you system into a precarious state where any new package installed or updated (example our pacman -S firefox command form earlier) will be a version that is mismatched with the rest of your system. This is unsupported and will occasionally cause problems. Generally speaking you shouldn’t run pacman -Sy, any time you are using -Sy you should also be passing -u. This ensures that the package list and your installed packages are updated together.

No, pacman -S firefox will not update your firefox.

pacman -Sy firefox will update your firefox and nothing else.

If you have done pacman -Sy once, then your list of packages and their versions gets updated.
From then on, using pacman -S <package> on any package, whether or not it was already installed, will now get the new version of it.

On the other hand, if you have not updated for long, then if you run pacman -Su to update, it will update nothing, because it looks at the old package list and compares it to installed packages and all of them match.
If you were to use pacman -Sy and then pacman -Su, then it would do the update, similar to pacman -Syu.
If you did pacman -Sy yesterday and then do pacman -Su today, then it will update up to yesterday’s packages and will ignore any updates from that point to today.

This can be considered analogous to apt update and apt upgrade.
If you run apt upgrade without apt update, you only upgrade upto the packages that you got until the last apt update.

If arch used apt, then in this case, the recommendation would be for never use apt update without using apt upgrade right after it.

You’re correct partial upgrades are unsupported. Arch follows a rolling release model, meaning there are no fixed “versions” of the system. Instead, everything is continuously updated. Each package is built and tested against the current state of the rest of the system in the Arch repositories. That package was compiled against the latest system libraries in the repos, not necessarily the ones on your machine.

Your proposed “workaround” may work if the package is standalone and has few/no dependencies. Again, ArchLinux strongly recommends full system upgrades (pacman -Syu) rather than only reinstalling/upgrading a single package, because library or dependency mismatches can occur if your system is out of sync.
A safer approach may be to use “pacman -S package --needed” which will avoid removing it first and automatically handles dependencies safely.

I haven’t used Arch in a decade, but as unstable as Arch is, I don’t think Arch doesn’t have dependencies poorly defined like that.

Say, firefox-1 does not depend on libssl-1, but now you upgrade it to firefox-2, you won’t succeed if you successfully downloaded firefox-2, but failed to download libssl-1, because pacman shall fail, while saying the reason being failed to download all of its dependencies.

If you start with a system with both firefox-1 and libssl-1 installed, upgrading firefox-1 to firefox-2 sure would have no problem, because its dependencies are already fulfilled.

If your system is breaking, it’s probably due to some other issue, but it could not be pacman’s.

It seems like there might be exceptions to the “no partial upgrades” which has not been discussed: you can pin your version of the kernel primarily to give time for packages like zfs to catch up to the latest kernel

When installing on Arch (or a derivative) you should run sudo pacman -Syu package_name so it is always up to date.