UPDATE
Good news and bad news:
The good news is that I was able to find matching female connectors and successfully spliced those into place.

The bad news is that the male connector doesn't pop out, but rather only pops in. I managed to clean the pins with a combination DeOxit and 800 grit sandpaper maneuvered with tweezers. I still have a low beam out on the driver side (as opposed to only having the passenger side high beam being the only functional light), so small victory.

I think it's time to call in a professional, as I feel I have done everything I can on my end. I suspect the fact that the driver side assembly had about a cup of water (I noticed it when I initially pulled the assembly just to replace the headlight bulbs) may have something to do with the issue. I'll likely end up in a situation where I'll have to weigh the price of an auto electric shop fixing it, or sourcing a used headlight assembly.

Thanks for all your input and coming along this adventure!