The belief in it, along with - as pp said - those who came after building up a power behind that believe has had a significant impact. Whether it's 'central' is subjective and unneeded. We can make a lot of things 'central' if we treat history like a story plot.
Judaism, we have Messanic Jews...but many still do not believe that Jesus will redeem them or yet, it seems many think it will be David who will redeem them. Scripture doesn't indicate this.
I've never heard any Jewish claim that the Messiah will be David. The qualifier is that a man from the line of David through Solomon. That is indicated in Jewish texts, that's why the Christian texts have two different genealogies of Jesus trying to tie him into that line, one in Matthew, One in Luke.
The other qualifiers are following the Torah, compelling others people to walk in the ways of the Torah through his actions, Fight the wars of God and win, restore of Temple in Jerusalem and gather in the Diaspora. There have been several potentials throughout history (in some branches of Judaism, it is seen that there are multiple potential messiahs in every generation), but if they die before completing it, they're disqualified, some groups phrasing it as failed such as discussions around Bar Kochba, others a bit harsher and saying they never were to start with.
There have been multiple 'potentials' who have died and then had people claiming they will come back, Menachem Mendel Schneerson being the most recent.
The entire idea of a Messiah is an exilic concept, it shows up in no texts prior to that, just like the concept of monotheism rather than henotheism or the Divine Council model common in the region at the time starts showing up in texts written or edited during or after King Josiah who, like Egypt's Pharaoh Akhenaten, made sweeping brutal reforms for his ideology.
Without Christianity we’d still be sacrificing children to get a good harvest.
With how much child labour is still involved in agriculture, I wouldn't say we've entirely moved past that.
Also, there are still areas of Christendom who argue that natural catastrophes are because of an unpleased deity, and argue murdering people as a means of solving the issue, some successfully.
And democracy - the Greek concept? The system the West has used in politics for a century-ish? How has Christianity opposed that?
Christianity, like all faiths, is very internally diverse, even more so in the early churches before Constantine got involved and things moved into what was orthodoxy and what was heretical, making it easier to control. From the limited records we have, the concept of Jesus as not divine was a norm in many early churches, that's only really come back into any form of accepted doctrine in any Christian branch within the last half century or so
Many Christians supported the Divine Rule of Kings and were against moves away from that, just like there were Christians who were against the monarchy.
The American evangelical churches pre-WW2 were anti-politics, they didn't get involved, it was too worldly. Those in power in those churches got invested into it in the decades after, and now...welll... yes, they've been very anti-science and anti-law when it suits them. There are still American and other Christians who support science and the law to an extent - I mean, it's not uncommon in some Christian spaces to see the phrase 'Jesus flipped tables for less than this' to discuss corruption in governments and other institutions, including church institution.
Christianity in itself isn't for or against anything and like all faiths changes over time and place. Those in power in Christianity, like any institution, it's largely been what benefits them and have used their influences to have their followers be for or against anything. The Bible, being a diverse document built over a century and with translation choices, it's not difficult for those in power to have the text be for or against most things.