Not Vector-specific – the same text is shown in every skin we have deployed. See Skin::lastModified().

https://hu.wikipedia.org/wiki/Kezdőlap?uselang=en&useskin=cologneblue
https://hu.wikipedia.org/wiki/Kezdőlap?uselang=en&useskin=modern
https://hu.wikipedia.org/wiki/Kezdőlap?uselang=en&useskin=monobook

I think for most pages, edits to templates are not considered "significant". If the infobox template is modified today, but the article about Star Wars on a given wiki was last edited 2 years ago, the footer should say it was last edited 2 years ago. And not reflect "today" based on Infobox_television having been edited recently.

The Main Page I suspect is the most common and possibly only notable exception where the vast majority of "content" is itself a template. I believe this is unusual and not what should dictate how the UI behaves in general.

Perhaps we could omit this by default in core by letting Skin::lastModified() return empty if the context title isMainPage(). There may be other cases where it isn't useful either, such as template namespaces perhaps.

Overall though I think it is useful in its current form. Requiring users to look at the history page would imho be a step backwards in usability. It is bad enough as it is that critical information that editors or readers can make use of to inform how they interpret or change content is often not known to them to exist or hard to discover.

Maybe mention both information somehow? E.g.:
This page was last edited on 12 December 2019, at 20:04, and some of its templates were edited on 20 April 2020, at 20:04.
or
This page was last edited directly on 12 December 2019, at 20:04, but it might show more recent content from other pages.

I also think the current form is the useful and tbh the correct form. The date should reflect the specific wiki page, not things it transcludes. If anything, an additional context e.g. "using templates modified on X date" would be useful, but likely a lot of work.