Christianity didn't exist until after the death of Jesus in 33ad, before that, it was technically a sect.
Jesus probably died in AD 30, although the only real certainty is that it was between AD 26 and AD 36 (when Pilate was around).
If Rome had been destroyed in the time of Augustus, then there would have been no Jewish revolt in the 60s and the Temple would not have been destroyed in AD 70. Judaism would have evolved in a completely different way from how it did and Christianity would probably never have existed - or, if it did exist, it would also do so in a very different way, perhaps remaining within Judaism.
However, Jesus by bloodline had a claim to the throne of Israel.
That's just later myth-making. No-one in Jesus' day could possibly have had any verifiable claim to be a descendant of King David, a figure of semi-legend who lived over a thousand years earlier. As far as I know, is there any reason to suppose that Jesus or anyone in his lifetime even made such a claim. When the later Gospel writers and others made this claim, it was simply part of their wider theological claim about who Jesus was and where he fitted into God's plan. Making Jesus a descendant of David says something significant about him and about his relationship to God, just as drawing parallels with Moses (as Matthew also does - eg the story of the massacre of the innocents and the flight to Egypt) says something else significant too.
Remember, the Jews who "killed" Jesus did so just as much out of fear of the Romans' response to a "rightful" king of Israel (Herod was pretty much installed by the Romans, he was not of the family of David, as Jesus was), so if they didn't have the wrath of their Roman occupiers to fear, the Jewish community might have better accepted Jesus as a leader, assuming Herod didn't track him down before then.
The Jews didn't kill Jesus, the Romans did, and they did so because he was a trouble-maker during the very volatile period of Passover. As I said, there's no reason to suppose that Jesus was really "of the house of David", at least no more than anyone else - it's like someone today claiming to be "of the house of Charlemagne"!
Also, the Romans weren't "occupiers" of Israel. There were only about 3,000 troops in the entire country and they spent almost all their time in Neocaesarea, on the coast. In the time of Jesus the place was run by the tetrarchs or, in the case of Jerusalem, the high priest.
The Herodians did suffer from some unpopularity in virtue of not being quite properly Jewish, at least in the eyes of some of their subjects, but I think on the whole this was low-level muttering rather than the cause of serious discontent - rather like people complaining that Gordon Brown is Scottish. Like most politicians, the Herodians were generally judged on the basis of their actions, not their background. Thus, for example, Antipas ruled more or less OK and enjoyed the relative approval of his people, while Archelaus proved to be very bad and had to be replaced. So I don't see any reason why the people might have particularly wanted to chuck the Herodians out in favour of any supposed descendant of David.