I think part of the problem with Jews not recognizing the Messiah is they were expecting all of the prophecies to be fulfilled by the same person—AT THE SAME PLACE IN TIME.
The Messiah did indeed come and suffered for the many as was foretold. He will come AGAIN and fulfill the conquering king prophecies.
This was indeed its claim. It claimed for fifteen hundred years--and the ancient versions of chr*stianity still claim--that all messianic prophecies were fulfilled at that one time, but fulfilled "spiritually" rather than literally. The notion that J*sus fulfilled only part of the prophecies (the literal ones) and will one day fulfill the rest of them in the future is a Protestant innovation (please note that I am not saying that "the second coming" is a Protestant innovation; rather the idea that the "conquering king" prophecies will literally be fulfilled in the future instead of already being fulfilled "spiritually" in the past is a Protestant innovation).
The Messiah did indeed come and suffered for the many as was foretold. He will come AGAIN and fulfill the conquering king prophecies.
First, until all the prophecies are fulfilled there is no messiah. If J*sus hasn't fulfilled all the prophecies yet, then he isn't the messiah yet. Second, how do you know he fulfilled the "first" part of the prophecies? All you can do is reason in circles: "he's the messiah because he said so and he's the messiah so he knows what he's talking about."
J*sus "fulfills" chr*stianity's unique (and erroneous) interpretation of the prophecies, not the prophecies themselves. And you continue to miss a very important point: one doesn't begin with the "new testament" or the Prophecies but with the Torah. Does the Torah authorize any such thing as chr*stianity? And "yes it does because the new testament says so" isn't an answer.