I actually thought that it would be helpful to show something in-app which I spun out from another discussion:
so that users are made aware that their instance is out of date - could even incorporate it with the Marketplace to also check plugins.
Maybe a dashboard widget with status of Core & plugins might be helpful?
In Joomla this is how it is done and you naturally know that is where to look - there is also a section for post-update notifications which shows useful information that might relate to new features or important release information for example.
With Joomla 4 it has now moved to a dashboard widget (their equivalent) which I find much more elegant (and easier to switch off, if relevant!)