Hi -- Just in case someone out there struggles with this issue, I wanted to share.
It was very simple: The latch assembly's bottom 10mm bolt -- there are three total -- attaches to a support rod/beam that, in turn, is bolted to the car's frame. If you peek through the grill or directly under the bumper it is very easy to see. My bolt was snapped! I'll drill it out or jbweld the thing later.
Anyways, without this bolt the entire hood latch flexes downwards and prevents the striker bar from having enough depth to properly latch when you close the hood. I used my car jack -- after stupidly trying a cooking book, -- to shift it up a mere half-bolt's length and hold it there while I then successfully closed the hood.
For those where this is not the issue and the problem is likely an internal spring, it is not hard to remove everything and see for yourself. I cleaned mine and put some grease in there which also helped, but was most definitely NOT the culprit. For those ordering replacement latches, I'm fairly certain a same-generation/year Corolla latch will work just as well and save you a good pocket-full of change.