Sometimes it's what we know for sure that just ain't so that gets us into trouble.

When the manual says A300/310 will not capture VNAV path from above...the meaning is you have to take some action to effectively get it onto the path. This most commonly occurs when you are descending to an altitude w/VNAV armed and fail to notice that the VNAV armed goes away after capturing that altitude. Then pass a fix and think why in the heck aren't we descending...agh, command a descent and arm VNAV again and VNAV will capture the path-eventually.

I maintain that this crew was NOT doing a dive\drive because there was apparently absolutely no awareness of the stepdown fix or minimums. The expectation was that the airplane would capture the VNAV path and the wx wasn't even close to mins so there just wasn't a discussion of crossing FAF 200' high, crossing stepdown fix on altitude. This crew knew that the wx was VMC (Something that was wrong. Wx was "filtered" and no pilot at UPS knew about that filter existed for FPR and ACARs Wx) The Capt knew ATC had kept them higher than he preferred and knew he was way high on the approach. (Which was wrong, crossed the FAF approx 200' high and in fact was on the VNAV path about halfway into the final segment.) The Crew knew they had set up the LOC VNAV approach. (Which was wrong, they both failed to catch that the FMS had not been sequenced appropriately for the approach. Information that it wasn't setup correctly was available in several spots. Failed to notice the next waypoint displayed was still KBHM. Failed to notice the VNAV path indicator indicated well below optimum path (because airplane thought they were flying to KBHM and then flying back out to the IF and then flying the approach). And failed to notice the erroneous data on the VNAV Approach FMS page (which is how you command airspeed on final while in VNAV on the Airbus. That page also provides a numeric indicating relative position to the VNAV path)

So, knowing VNAV did not intercept because he was exceptionally high. The Capt commanded V/S to descend and intercept the VNAV path. He did so without communicating that fact to his FO. And he commanded a descent speed much greater than I would have done in a similar situation. They hit 1000' AFL and were unstable and still in the clouds. IMO-the Capt was the only one truly aware he was descending at 1500 fpm, and continued to do so.

Blew through approach minimums without the rwy in sight, with no apparent awareness on either side that they had done so...although they had mentioned minimums 24s prior to crashing. And when mentioned, they were approx 200' above minimums and continued to descend at 1500 fpm.
They were 90' below minimums when they both commented about the approach having to be an actual IMC approach.

IF the Capt was intentionally doing a dive\drive he should have zero'd out the V/S approaching minimums.

Airbus doesn't have an upgraded FMS system. Minimums are set by an altimeter baro bug. Missed approach altitude is set into the MCP, usually by 1000' AFL.

Same is true for ILS approaches.
There are ILS approaches where G/S is not supposed to be used below mins, but I'd doubt the MCP is set to ILS mins to "protect" you from utilizing the G/S below those mins. Instead, the PF is supposed to be paying attention and see the runway or go around. And the PM is supposed to be paying attention, and if need be command a go around