DaTruf
You certainly could do that. Might help keep it clear.
The issue here is that the port on the kettle/pot/keggle is usually above where a false bottom would sit. If this is not clear than false bottoms require a dip tube to go from the hole in the side of the pot, through a hole in the false bottom and picks up liquid below the false bottom.
Logistically It doesnt really make sense, because the dip tube with the false bottom helps to minimize any liquid that otherwise would be left behind. A bazooka screen usually leaves a bunch of liquid behind, unless you do an potentially scary/heavy tip or dump!
If you really want better filtration that a false bottom would allow (which is pretty good) some folks put a stainless steel kitchen scrubby/brillo right where the dip tube picks up. Other folks use silicone tubing cut horizontally around the perimeter of the false bottom, where the FB meets the kettle walls, to ensure a better seal.