As Jamie mentioned above Firefox is trying to launch a new process and is unable to find a matching binary. Before Mozilla detected the in-between update state there it was common to see spikes of crashes that were related to process spawning.
The choice Mozilla has is to allow the user randomly crash with a chance of profile corruption or alerting the user about the state and asking the user to trigger an orderly shutdown.
Power users can disable it. For the less advanced users, impose that as default behavior as its pros and cons.
There must be some additional context here. I've never observed this behavior on Windows or on Linux (although on Linux I'm using an LTS release, which could be a confounding factor).
I do applaud firefox for supporting so much customization, as in this case. If I weren't able to disable this behavior I would have stopped using firefox immediately.
Obviously this is only a problem on Linux, because on other platforms Firefox's own autoupdater will only apply the update when Firefox is restarting.
In my experience, running software updates while using the system is a pretty bad idea in general. It works well most of the time, but there's a reason Linux* is the only system that allows that (every other OS defers applying updates to a reboot).
It happens because Firefox closes and re-opens libraries and permits an in-place update (and thus, breaking itself).
While I understand that it is very annoying if you hit this issue, it is not done as some kind of trick to try to get you to update. The issue is that Firefox is trying to start a new content process to load your page, but the old executable no longer exists so it is impossible.
I don't use Windows much these days, but I'd love to know which setting it is that turns this off as I tend to hoard tabs and this sounds extremely annoying.