What I don’t understand is why would anybody be weighted to the point that they would not be able to float on the surface with or without a full tank of air?
I always thought a diver was supposed to weight themselves so that they have to swim down, not drop like a rock as soon as any air is dumped from the bladder. Am I missing something?
At the start of the dive (assuming an 80CF tank) you are carrying 5lb more weight than you will at the end - the air that you are going to breathe out of that tank (0.08 lb/CF).
If you can float at the start of the dive, and "swim down" until wetsuit compression makes you less buoyant, then at the end of the dive you will be whatever your starting buoyancy was, plus an additional five pounds of buoyancy.
As has been pointed out above, that will make it challenging to hold a safety stop, not to mention whatever shallow exploration you planned at the end.
I weight my students at eye level with a normal breath, and an empty BCD. Then we look at their tank contents. If they checked with the classic 500-psi tank, then we just change tanks and go diving. But if they have a full tank, we add whatever weight of air they will breathe off during the dive, which means they are a few pounds heavy at the beginning of the dive, and just a little buoyant at the surface at the end of the dive, which means they should be neutral at the 15-foot stop at the end.
At least that's my technique.
Now, all that said, any single tank diver with fins (even though he/she isn't "floating" at the beginning of the dive), should easily be able to "stay afloat" despite loss of BCD function. And if you drop weights carried because of your neoprene and the air you plan to breathe off, you should be able to float passively without much finning.