Do I need to worry about air bubbles in the first coats of drywall mud?

The purpose of sanding (or scraping) between coats is to knock off the high points. NOT bonding.

Mud (compound) fills the low points, but it shrinks, requiring multiple coats.

The goal is to have something close to a flat surface, which you get most easily by removing the high points and filling the low points.

If you don't remove the high points, you end up filling a larger area with more mud since the knife rides on the high points. More mud shrinks more, so there's more remaining low spot to fill with the next coat. If you knock off the high points, you are just filling the low points, and the amount you add each coat is less, so it shrinks less.

Sure but you still may have pits or birds eyes because of the build up and you will have more sanding to do. The method I recommend Pole or hand held sand block corse mesh or sand paper. First coat , when dry 1 rub up then down Second coat usually my final coat , same sand block up then down DONE , Your method you will have a bunch of mud to remove, waste of mud on extra coats and you have to sand more. After all that sanding more to clean up.

Not sanding between coats sounds ok but actually ends up using more mud that has to be removed more work more mess.

But you can do it that way I have seen diy take a week to tape and fill screws in a 10x 12 bedroom nothing fancy , that’s only a few hours work but they were happy and I am sure you will be also but your process waste time and $ in my opinion.