Christmas was obviously picked by the church for maximum impact
It's all part of the cultural assimilation done by the mid-period church (much like the concept of the Trinity, the idea of an immortal soul and a literal devil - all imports from pagan religions).
The process goes like this:
1. Send missionaries into an area and convert the rulers.
2. Once the rulers are converts, don't make them give up their usual festivals but re-brand them.
3. Over time the origianl source gets muddled and people forget.
s the timing of that is related to the Jewish Passover festival
That's because Jesus died at Pesach.