Got it working. Not sure which thing fixed it, I have about 3 theories.
All of my theories are based on the door handle linkage not fully releasing. The door wouldn't unlock because it was like someone was trying to open the door while unlocking it simultaneously.
Theory #1; I think the linkage pointed at by the green arrow is supposed to be all the way behind the sheet metal and popped up behind the sheet metal like it appears in the picture. I noticed a dimple in the door sheet metal that's apparently from this piece getting forced into the door panel when the door handle was pried off. I suspect the door is warped enough that this piece is not far enough into the door to pop up like it should. I pounded on the handle with a hammer on a wood block to try and pop the handle section of the door handle back in further.
Theory #2 is that the rod pointed at by the blue arrow was jammed up on the bracket it hangs from and wasn't dropping fully. Again the hammering I did would jar this loose.
Theory #3 is that the linkage is jacked up in general from the handle being torn from the door. Could be my messing with it managed to get things settled just in the right position to start working properly.
And maybe it's a combination of 2 or all 3 of the above.
The handle still isn't 100% right because the button doesn't not extend out fully. I'm planning to swap out the entire door at some point in the future. For now it's working again, that's the important thing.
I included the red arrow above because that's the roll pin I knocked out to swap the handles. It was easier than last time I swapped a handle using the clip on the inside of the door down by the door latch. Although maybe that's the reason I had trouble with the door unlocking and it's not a good idea to remove that pin.