How do priorities work for jobs?

The priority system is completely separate from the job board. The job board just increases stats and gives perks, it doesn't mean they'll prioritize the job.

Maybe this quote will help you to understand answer:

Errands can be prioritized 1-9 (1 is lowest priority, 9 is highest). High priority errands are always performed before low priority errands. However, a busy Duplicant will continue to work on their current errand until it's complete, even if a higher priority errand becomes available. source

Important: This is currently valid for Occupational Upgrade(v254439 and later).