'The Jews expected a warrior messiah who would be a king..' etc. etc.
Expected. Past tense.
Well, I think this is a really good question. As a modern person, my concept of Moshiach comes from Tanach. But I have access to the entire Tanach. It's kind of hard to imagine 1st Cent. Jews knowing Tanach in that way, or being educated the way I've been educated.
In the Gospels, if I recall, Jesus' rejection by the Jews was never, "You're not the Mochiach!" And Jesus only maybe-maybe claimed to be The Messiah 1 time in public and 1 time in private. Again, based on my memory of the story.
So it seems like, at least according to the Gospels, "Is or Is Not the Moshiach?" wasn't even a question. So I'm not sure how a claim is made about what was expected at that time.
However, the answer I think is going to be dIfferent if the Christian is claiming, "Hey you, modern person, you're expecting a warrior king, but it's not, because..."