What do you mean the "page's short description"? Something provided by TextExtracts ?

I mean short description, the line of text that is associated with a page in search results and other places. I have no idea what TextExtracts, but looking at its documentation, it doesn't seem to be what I'm asking for.

Ok, so it's a Wikidata thing

Probably via mw.ext.wikibase mw.wikibase then, rather than mw.title.

Just to be clear, I'm not trying to get the description located at wikidata, but the one at the local wiki. See this page information "Local description" section.

Short description handling currently lives in Wikibase, so mw.wikibase makes sense nevertheless.

This should be reasonably easy, there's a service to return the description, it just needs some scaffolding to make it available to Lua.

Just to repeat myself, again, mw.wikibase is completely irrelevant to this, as is Wikidata. I'm not trying to access the description at Wikidata, for that I can use mw.wikibase.getDescription, which is already available. I'm trying to access the local description which can be seen either on the article itself, or in the "Page information" page. That can give a completely different description.

You can claim all you want it's not Wikidata/Wikibase, but it is. Look at the message keys used

wikibase-pageinfo-description-local is defined and used by MediaWiki-extensions-WikibaseClient

Local descriptions are not touched by the Wikidata team so adding @EvanProdromou to get it on the right people's radar.