Firefox won't auto update

And after the "notification that an update failed to install and to install it manually" you downloaded the full installer to have a full binary reinstall — or just let the partial updates try for long enough on their own to eventually succeed?

It takes me to the Mozilla page to download the installer

Correct. And you used the full installer to upgrade fully already — instead of the in–app updater… and then the situation repeats in a couple weeks again?

Yes, that is correct

There's a semi–(un)related thread support.mozilla.org/questions/1587819 that nonetheless has a scheduler–specific advice to check if that might interfere in some way. Generally some previous wrinkles are captured in How to fix the Update Failed error message when updating Firefox, but the root cause could be very individual in that case (and would need investigating deeper).

I usually just advice folks with permission/access issues (as is the potential culprit here) to simply get Firefox from their Microsoft Store and let it handle the updates for them.

Depending on how much you're willing to tinker with the issue, it could be helpful to file a bug and assist the engineer with getting various troubleshooting info in case this might affect more people on regular basis — or if you're willing to poke at some of the updater internals like the data it pulls, where it's stored etc. if there's anything out of the ordinary… if that sounds like something you're comfortable with.