The way I’m used to it is: 1. Wake up 2. Breakfast 3. Brush teeth.
Having it as 1, 3, 2 doesn’t make sense for me.
The health related reasons others gave are interesting, but my reasoning is this pretty simple: On normal weekdays I don’t really have breakfast, so brushing my teeth as one if the first things makes sense. On weekends I like to have breakfast with my family so I’d like to get rid of morning breath before I talk to people.