It's a good bit, I admit.
I do want to note that in the Gospels and really basically everything remotely intended to portray "eye witness" events of regarding Jesus (still stories that copy heavily from one another, hardly multiple accounts), the main message is, over and over again, "Jesus is special, you must believe in him over everything else". Reading these texts, the message is not, "love each other and live in community", it is, "eschew all else if that is what it takes to believe in me, even if that means being kind to your enemies". Just about every parable is just iterations on the theme of, "Jesus is special, he does miracles, you need to believe over all else". The Gospels are written with the exoectation that this would all be wrapped up during their lifetimes, as in Jesus was coming back on Thursday so you better start believing and stop focusing on worldly things.
Which is to say, the figure of Jesus is not exactly socialist, as socialists focus on the material, on community, of creating a better world through the destruction of capitalism. The figure of Jesus is more like a hippie that says, "screw all of that, join my cult, nothing else matters".
Anyways do with this what you will. The "Jesus was socialist" angle is still decent agitprop because most self-proclaimed Christians haven't read The Gospels. Very few people are really engaging with the figure or reading critically, they just learn parables in isolation and a local religious figure tells them what to believe about it.
The Democratic Party cannot be taken over. It is not a democratic institution. You could have 70% of elected Democrats agreeing with you and being variations on Mamdani and "party leadership" wouod still be outside your hands and they would still be working against you and there would be nothing you could do to advance your position in the party. It is a private, bourgeois institutional governed by ita donors and those donors' supplicants.
More practically, what you could hope for is entryism and then a split. That is the basic claimed premise by DemSocs in DSA who argue for running as Dems. Unsurprisingly, this is a Trotskyist tactic and basically never works. But trying and failing in a very public way may still be a boon for radicalization and growing our ranks.
But only with discipline (or luck)! The Dems will try to coopt everything appealing about DemSocs without stepping on the toes of donors. And that cooption begins with Mamdani himself, who will be constantly pressured to soften his stances and be more lib. DSA has zero discipline whatsoever, so with someone like Mamdani one would have to depend on luck, i.e. just this one guy being principled. This is the DSA way and it is why they eat shit in 90% of their electoralism, they back candidates with no vetting and usually provide no support. But maybe they got lucky with this one guy.