I recall my early studies during my reversion process about how the Messianic concept evolved in Judaism.
When the Roman occupation occurred is when the personal Messiah concept developed. Due to the oppression from Rome, the Messiah was solidly linked with the Davidic dynasty when previously it was only infrequently connected therewith. The Messiah was at that point in time hoped to be a new king that would break the yoke of Rome and bring freedom to the state.
Jewish hopes had always linked the Messianic Age to come with "redemption," which in Jewish theology is limited to relief from injustice, poverty, sickness and other ailments of the physical world. The Christians reinterpreted this, likely for convenience, when Jesus of Nazareth ended up dying. The "redemption" now tied with the Messiah by Christians was one from "sin" inherited from Adam, a concept foreign to Judaism. The other physical facets of "redemption" were thus transferred to a future Second Coming since it was not possible to totally invent the Messianic concept without using the literal and basic meaning of the "redemption" which is still central to the Jews.
Today it is generally recognized that the coming Jewish Messiah who would be a monarch is not likely to be the case. Monarchies are institutions that have become obsolete, so what the Jews expect (while not necessarily rejecting the possible revival of the Davidic dynasty) has advanced beyond the ancient descriptions. Some Jews have moved beyond the concept of the Messiah even being a person, believing that the prophecies and hopes may have been using personifications and ancient concepts to describe something else, perhaps even something that humans can achieve together through humanitarian efforts.
This does not mean that Christians are lost either. The basic ideas of what the Messiah concept entails has now been spread abroad through the ministry of Christians to the extent that should things turn out different from what Christians expect, they should still be able to accept whatever the fulfillment of the Messianic hopes eventually turn out to be.