Sorry, no photos, I was already a bit sad about that. But it looked about the same as the photo above. The insulation on the oil breather continued to burn, and it was that which I could see burning, and extinguished with the fire extinguisher in through the oil door.
I took the carb apart myself, the engine shop then overhauled and certified it for me. The two piece venturi was blackened, and distorted enough that the engine would not have run with it that way. It was about ready to drop out of the three clips. Though not melted, it was evident that the inner venturi got hot enough to begin to melt, I could see it.
That is the reason for the AD to inspect the venturi's. A carb fire could have distorted the venturi a little, but the pilot got it running, and went flying unknowingly. The only way to know will be to look at it, and assure it's correctly in place. They have fallen out of position later, and they you're gliding.
The volume delivered is limited by the bore and stroke of the accelerator pump which are both small compared to the primer.
Check again.... I think you'll find that they have very similar volumes. The accelerator pump (of an MA-3 carb) is in the range of 5/8" diameter, and strokes about an inch. A Piper hand primer has a plunger diameter if less than half an inch, and about the same travel if I recall (it's been a few years since I flew a Piper. More importantly, the more slow movement of the hand primer is because it is making much greater pressure, and atomizing the fuel to a much more fine mist in the air. (so flow is much less). This mist will quickly evaporate into vapour (which is what burns - liquid does not burn well). The fuel vapour will hang in the induction air passages for some time. Compared to the stream of liquid gasoline (about like a water pistol). If the accelerator pump squirt is drawn up the induction by good airflow (running engine), it gets to the cylinders as intended. However, if the airflow is poor, and gravity good, the liquid fuel will just drop down through the carb, into the airbox, and pool there. Once the accelerator pump fuel has dropped into the airbox, it is of no use as priming fuel, and the start may fail. More priming will be necessary, which will probably get the engine running, but there's still the the pooled fuel. Only a while to evaporate it, or a running engine to draw it through will get rid of it. That'd be why getting the engine up to full cranking speed before introducing the full prime make some sense (and works well for my O-360). No matter what anyone says, a pool of fuel, where there is a possible source of ignition, is not as safe as no pool of fuel there.
In aviation (like many other things) . There are "other ways" to do things, some of which work - but the fact that they work does not mean they are the better way to do it! Noting Lycoming's own remark about the primer being the "best" method to start their engines.