The Falklands reinforced the need for a ship's gun

Did the lack of destroyer guns and the absence of cruisers matter after the Falklands War?

Yes; the experience helped end the Royal Navy's brief move toward major ships without a main deck gun.

The absence of cruisers and the lack of main guns on some ships both became issues after the Falklands. Early Type 22 frigates are the example of ships built without a main gun. Other ships had lost their guns during modification. That experience is linked to the subsequent retention of a main deck gun in Royal Navy designs.