It's good enough but you're probably not gonna be impressed if you play FPS's all the time. Other people have pointed out already that the irony is that you have to be interested enough in military FPS games to even get into the game enough to see and get the subversive part. Which kinda works against it on two levels, because people who are into War shooters enough to play it without knowing the true point of the game probably didn't appreciate being called out by it, and people who would appreciate the subversion probably aren't the people who would pick the game up to begin with without knowing there was something different about it.
It's mentioned in the video that Yahtzee may have done a lot to get people aware that this was something a bit different then a lot of the other Brown Modern War Shooters all over the place at the time, though I have no idea how accurate that is. Presumably someone at Jaeger certainly thought so.
AFAIK both Vanquish and Spec Ops are always TPS, unless they pull one of those Deus Ex kinda hybrids. But yeah, FPS has become a rather synonymous term with pretty much anything shooter-oriented in recent years similar to Soulslikes being a catch-all.