I've read David Mack's Desperate Hours prequel set on the Shenzhou. I also think the Saru-centric book (Fear Itself) is also set on the Shenzhou or at least is a prequel to the Discovery pilot (I haven't read Fear Itself so I can't say it is set on the Shenzhou for certain though). There is also a Georgiou & Lorca novel (Drastic Measures) set almost a decade before Desperate Hours, though I don't know if that one features the Shenzhou either.
If you do incorporate Desperate Hours, Mack has done a lot of the groundwork already, establishing who the first officer was prior to Burnham. Also, Mack did some nice sketches of characters, as well as naming characters whose names were made canon on the series.
All that being said, I had a similar thought when I watched "The Vulcan Hello" and wanted to see more of this crew. I also thought about it when I was reading Desperate Hours, which I finished long after seeing the pilot episode.
I don't want to commit when I'm not sure I can deliver at the moment, but put me down as intrigued and a maybe.
Not sure how long you envision this. If it is a Season Zero, then it would be Year Six of Burnham's time on Shenzhou and leading into "The Vulcan Hello". Desperate Hours takes place one year before the pilot episode.
https://memory-beta.fandom.com/wiki/Desperate_Hours
https://memory-alpha.fandom.com/wiki/Desperate_Hours
https://memory-beta.fandom.com/wiki/Drastic_Measures
https://memory-beta.fandom.com/wiki/Fear_Itself
https://memory-beta.fandom.com/wiki/USS_Shenzhou
Perhaps there's some inspiration from the "Age of Discovery" part of Star Trek Online:
https://sto.gamepedia.com/Age_of_Discovery