Why would Captain Rayner demote himself to a measly First Officer for Burnham?

He did not demote himself. He was demoted for his actions by a review board. They showed the board in action. Afterwards, Burnham requested he serve as her first officer.

Author of the Sodality Universe
The Road from Antioch
In the Markets of Tyre
Flight to Lystra
The Theater at Ephesus
The Council on Jerusalem

The Shattered World begins in 2025

He basically looks like an old white guy and understands military discipline and order and getting things done vs. spending an hour talking about feelings before making a decision, so he had to be humbled and proven wrong and schooled by a young woman of color. This becomes explicit in the 4th episode, just in case anyone was dull enough to miss it previously. I'm as liberal and "woke" as it gets, but even I think this is the worst kind of drivel.