Yes, the wording makes it agnostic to what kind of action is used to cast the cantrip. The design choice is a little bit of a head scratcher, since casting a cantrip almost always requires some kind of action, but then Sage Advice swoops in and says that was the intent. Speaking personally, the result is far from overpowered, so I wouldn't have any issue with it regardless.except War Magic does not say "you use the action to cast a spell". It says "use your action to cast a cantrip". And when you use the bladsinger extra attack feature, you are 'using your action' and 'you are casting a cantrip' and thus like i said, the wording is easily read in a way that the synergy is perfectly compatible, so i understand why Crawford ruled it as compatible.