It doesn't open during the cycle, and it's pretty hard to open it manually.
Nothing happens when I push the dark grey button. Also, I'm supposed to be able to use a toothpick and press a light grey lever sideways--nothing happens then, either. The only way to open the tray is to lift the tray front slightly and pull the tray back. I did take the lid off and check for gunk--the tray and the slots were quite clean.

What should I explore next?

I can't tell if they are warped. Nothing obvious.

The old thread is for a different model with a different kind of detergent dispenser.

There's a U-tube video showing how to pop the cover off and clean the dispenser--and that's the same kind I have. However, it shows how to move the one lever--it's supposed to trigger the door to open--and for the life of me, I can't find any part inside the door that the lever should contact.

One thing I do notice. There's a parts diagram here: at c(dot) Sears Parts Direct (dot)com(slash)lis_png(slash)PLDM(slash)50044418-00002(dot)png (remove spaces and replace items in parentheses)
and I notice that the spring for the door on my machine is shaped differently from the spring shown for part #234 in the diagram. In the diagram, the two ends of the spring are roughly 0 degrees, and mine is closer to 90 degrees, perhaps a bit more. (I wish the diagram would show the opposite view of the one piece in the diagram that does not have an individual part number--I could then possibly draw arrows to the things I've referred to.)
I've seen options to order a new lid and spring only, or the entire assembly. I'm pretty handy at putting things like this together, especially if I have clear directions, but it sure would be nice to figure out in advance if I only need the spring, or if I'd need the entire assembly.