You’re right. It is the job of the surface navy to show the flag. Sub captains never, ever want to be detected. Their idea of serving notice that they are in the area is when your sonar picks up an inbound torpedo. Other than that, you just never know.
The surface fleet of the RN is pretty stretched as it is. And to make a statement, you would practically need to send at least a couple of frigates/destroyers and supply vessels.
Maybe the demand to send a sub is to warn the Argentines of consequences of any aggression. A single Astute class sub has enough firepower to sink most of their surface navy and shoot off a few Tomahawks without them even knowing.