There should not be much, if any, air showing up in the brine tube during the fill operation. The presence of continuing air suggests a problem in the head (unless you have lots of air in your water supply). Also, the fact that you only got 3 gallons with the BLFC removed suggests a problem with the head or air in your water supply. The fact that the unit seems to adequately draw brine suggests there is no obstruction in the brine line itself. You should contact the seller of the unit--there is a 5 year warranty on new Fleck Valves.
A couple of other thoughts--
BLFC buttons come in many sizes, including .125 gpm. The fill rate you are reporting is consistent with a .125 gpm button.
Usually there is a sticker on the unit that indicates the size of the BLFC button in gpm. Is there one on your unit and what does it say?
Have you checked with the seller of the unit to ask what size BLFC is installed?
Who did you buy the unit from?