But did the Romans care for his movement in the first place? I mean, before his death
Its a hard question to know. The Bible shows Pontius Pilate as extremely reluctant to Kill Jesus. If he acts on a desire to end the trouble without violence, exiling Jesus is probably the right thing to do--and I'd suppose this would throw a large monkey wrench in everything.
I see it at least possible that if the situation in Judea became provocative, Pilate could respond with means other than execution--he did attempt the crown of thorns and stuff, at least according to the bible.
Now, I don't consider the Bible a factual source--it certainly does not have primary source in any case. There is room for doubt that Jesus was even a historical person--and it is on this point that I will concede this, temporarily, for the sake of this thread. We don't have much else to go on, and we are not helped by a millennium of religious leaders turning Christianity into a propaganda machine at the expense of accuracy.
So, the point is that Pilate quietly kicks Jesus out, and Pilate essentially calls this Rome's word in the matter. Jesus' Cult would either be stillborn or gravely marginalized as a result of these actions. Making it popular enough to reach Rome and become its primary faith is a non-issue.