I would like it to be flush with the wall

Or maybe a ring pull. I have a mirror on drawer slides so it opens parallel to the wall. Nice, creative and functional solution!

Whqt about a ”push to open” type mechanism? No hardware, and you can operate it with the back of your hand.

I would think:

A skilled craftsman would be needed to properly construct such a detail.

Over time wear spots would be created that would require maintenance, especially with a ”push to open” type mechanism.

It reminds me of an access panel in a commercial building.

I have the cutout in the wall. Wondering how to have a hinge to open the drywall

I did this, The contractor had to make a box that fit into the framing. sat flush with the drywall and was able to be mudded into the drywall to sit completely flush with the wall. The door is hinged with concealed hinges, like you would used on a kitchen cabinet. My door has a small knob instead of an edge pull, but an edge pull would have been fine. It is covered with wallpaper to match the wall.

I do not like push to open mechanisms because they often get out of whack and don't sit flush with the wall and doors with that type of mechanism do get wear marks or burnish marks from touching them all the time and you don't really want this.

Comparing to hanging a recessed medicine cabinet that is prefab (hours) this was a detail that took days to execute spread throughout a week or more because of finishing, spackling, sanding and so forth.

I was hoping that wasn't the case 😂

The wallpaper can cover-up a lot.

No, it can't. Wallpaper on a flawed surface looks like crap.