Above is a video that I saw on Cdr Salamander’s “Full Bore Friday.” This happened about 43 years ago so there is a good chance you may not know about it. The British lost several ships but still won the war. Ironically if Argentina had waited for a few months some of the critical ships would have been decommissioned, most importantly the two HMS Fearless Landing Platform Docks. Instead they remained in commission until 1999 and 2002.
The HMS Coventry was slightly smaller than a Nation Security Cutter or an Offshore Patrol Cutter. During the fighting the Type 42 had no close in weapon system but they were added shortly after the fighting ended.
It was a failure to add what was needed.
Chuck Hill's CG Blog 
The RN’s air defence efforts at the time were directed towards taking pot shots at Bears mid-Atlantic so Sea Dart was enough. Supposedly.
Mass was important too. There were 12 Type 42.
Not to hard to imagine how the Falklands War would have gone if Type 42 had resembled the Italian Audace-class destroyer. But then again the Italians could only afford 2 of those.
Since Phalanx was only approved for production in 1978 it probably would have been impossible to get them Phalanx before going into battle, but I remember thinking that they could really have used it.
Right now I feel like SeaRAM is the absolute minimum