I would simply paint the door to the living area completely. What’s the point of it? It doesn’t really separate anything since the staircase doesn’t even branch off from the hallway. So you get noise from the living area to the bedrooms (and vice versa) anyway, which is actually the typical argument in favor of a door to the hallway.
A stroller will be super difficult anyway, the hallway is narrow and the setup with the two doors is annoying anyway. To squeeze the stroller in there you have to maneuver it carefully.
But I wouldn’t build real estate specifically to accommodate strollers either. You need a stroller (per child) for 1-1.5 years, then another buggy, and by 3, at the latest 4, it’s gone.