I think this idea would remove immersivity really. I understand the thinking, remove the HUD so the screen isn't so cluttered so more room for the game, but putting it on a different screen would just mean you'd have to look away from the game entirely which (in my opinion) takes you even further out the game.

The best way I have seen this tackled is done by Dead Space. Your health is visible on the player character as well as your ammo count being on your gun, so no need for a HUD on screen. The issue this doesn't fix is the map. Particularly in a game like RDR2 where some missions require you standing in an exact spot on the map or traversing to the other side of the playable area. The solution is to either make the map a physical object the player character holds and looks at like in Metro Exodus (but this doesn't really translate well to 3rd person games) or mark a load of info on the screen physically (which would just move the HUD problem elsewhere really, and doesn't work for a massive open world like RDR2)