My mom has a house with a central return and as you suggest her doors have about a 1 5 gap underneath to allow air to flow.
Gap under bathroom door for ventilation.
There should be plenty of room under the bottom of the bathroom door to allow air to get into the room so the fan has a source from which to pull the air it needs.
Client always resist 1 gap over the floor covering.
And for proper bathroom ventilation your exhaust fan needs to vent outside not into the attic or another area of the house.
There is a need for an air gap at the bottom for ventilation.
I like the suggestions for wall ceiling jump ducts etc.
If more gap is required it is easier to trim a door than to lengthen it.
If the bathroom vent can t get any air of course it isn t going to work the right way.
However the size of that gap is very unlikely to be greater than 1 2 for any door.
Using the link you have provided it has a picture of a french full lite door with a vent that has removed a significant part of the door s base.
I have a problem with the door vents you are passively promoting.
Check your bathroom door.
The flow dropped from an unrestricted 150 160 cfm down to under 100 cfm in each room.
I think it is too big and think a 1 4 1 2 gap would accommodate any unevenness in our floors and allow us to use area rugs.
There is an approximately 1 5 gap at the bottom of the doors.
If your door has very little or no clearance at the bottom you.
This is less than desirable in bathrooms and bedrooms.