Why did Jesus die during Passover, not Yom Kippur?

Edit: To move the bottom line up front, I am primarily interested in historical writings from theologians on this thing specifically, not just your understanding of the Gospel.

Passover is the central Jewish holiday, celebrating the story of how God delivered the Israelite people from Egypt, and specifically from the 10th plague.

The passover sacrifice and Jesus as the Paschal Lamb have become inseparable in Christianity. And this is sensible, of course - salvation, and the timing of Jesus' death.

Even though the Exodus did not happen and is most likely a story about the ending of the Exile, Passover is about the temporal salvation of the Israelites. It is not about sin.

We are told, though, that Jesus' death was a sacrifice to atone for our sins. This is a problem, though - that's not what Passover is about. Yom Kippur is the Day of Atonement, and is about 6 months later in the year.

Why would the Gospel, Jesus' death as atonement for our sins, happen on a holiday that has nothing to do with our sins whatsoever?

I'm sure this has been written about over the years by Christian and Jewish theologians. I'd be interested in reading their thoughts.