I agree that even in a potential conflict against China ASW will be a (the?) primary role for the RCN. A key objective for China in a war will be to keep US (and allied) forces as far away from Taiwan as possible and if possible to prevent shipments of key American materiel from reaching the war zone. Our CSC's would play a very important role in keeping Chinese subs away from US assets.
Yes there will also be missile attacks on Western forces but the USN has ships specifically designed to counter those attacks. If we feel that we need to up our AAW capability then as
@Furniture says we should get a dedicated AAW warship (highly unlikely due to money/personnel constraints) or look at other ways of increasing our missile capacity - most likely through minimally/unmanned arsenal ships - rather than tinkering with the CSC design.
$0.02