I think Israeli society is more diverse than that. Do Soviet jews have a big sense of unity with Mizrahim? Why is this different from any European country, where share of immigrants is probably lower than in Israel?
Once they are in Israel, probably. They are all members of a tiny world minority in their homeland surrounded by perceived enemies. That likely forms strong bonds plus they all serve in the military.
I don’t think that Jewish organizations like Birthright discriminate. They want all Jews to have children. I can’t think of anything equivalent to Birthright that operates on a national level.
How Birthright, a non-profit that organizes travel to Israel for young adults from other countries[1], is relevant to Israeli demographics? On their scale of operation, even if they had a significant repatriation rate (which I doubt), it would probably be just a 1-2% of their target group.