Your hypothetical setup is actually what I do normally. I routinely put together my gear when I get on a boat and turn on the air for a few breaths to make sure it all works and i don't have any o-ring problems, before turning the air off again in case of leakage during a long boat ride. I've made it a habit to look at my SPG when taking a few test breaths just before entry. Even if the breaths seem normal, the needle will dive towards zero, indicating that you've forgotten to turn on the air.
The air remaining in a hose in this scenario is good for one normal breath and then maybe one very hard breath as the pressure plummets toward ambient. Assuming that you are doing a "normal" entry, you'll stick your reg in your mouth before jumping in and realize by the second breath that you have a problem. Certainly you would realize it by the time you finished your entry. The worst case of this scenario is if you were doing a hot entry with an empty BC with the intention of kicking straight down fast, like is done for dives directly onto Peleliu Corner, but even then you'd be at maybe 10 feet before realizing that you can't draw a third breath.
At the risk of putting numbers on things, let's think through my weighting in this worst case scenario. You get a 10 pound buoyancy spread using just your lungs. My AL80 loses 6 pounds over the course of my dive and I never touch my inflator during a normal dive. Since I can still descend at 500 psi by completely exhaling, let's say that the 6 pound range runs from the 0# to 6# range of my 10# lung range, so a 3# breath (or two-thirds capacity breath) is "neutral" for me at the start of dive. I take my first breath on entry and exhale completely to descend (at this point being -7 pounds) and then start my second breath, realizing that there is problem. (Half breath so I'm at -2.) If I'm very quick thinking, I continue inhaling my last breath as I change direction and start heading up, blowing bubbles from a full set of lungs. (+3 pounds of buoyancy!) If I'm stupid and pause to figure out what is going on and do a normal exhalation to one third capacity in what is likely my last full breath on the reg, I'm at -4 pounds, which is worrying, but certainly not going to prevent me from finning up and getting to the surface, probably in a big hurry. If a diver doesn't think that they could fin back to the surface from that depth, I would agree that they are overweighted and should rethink their buoyancy.
Of course, some divers on SB will tell you that you should have drilled OOG routines so many times that you shouldn't even have to think about the fact your breath is coming hard before your right arm automatically crooks over your shoulder to turn the tank knob as you continue your descent. I only dream of being that hardcore.