First try resetting tool. With tool selected right click on tool icon in upper left corner.
I think the Magic Wand Tool is working correctly but you've missed the correct setting for what you want to do.
Looks like you have a white background layer and a semi-transparent layer that's been painted with a single colour, pure black I guess, at various opacities.
If the MWT is not set to "Sample All Layers" and is used on the semi-transparent black layer then it's not going to see the grays that result from blending the layer with the background. It's going to see each pixel as either fully-transparent or simply black. The behaviour which you do not want.
If the MWT is set to "Sample All Layers" then it will see the grays that result from blending the layers and the tolerance control will be effective. The behaviour which you do want.
Oh my gosh. I feel so sheepish right now because that is just such a simple fix and I've been freaking out this whole time trying complicated stuff - and this is the very simple solution!
Gosh, thank you. I was flipping out so much over this. I feel pretty dumb now! :C But thanks so much for pointing this out to me! I'm really thankful for your reply!
I think part of your difficulty in finding a solution was that resetting the tool or all Photoshop's settings will disable "Sample All Layers" and produce the behaviour which you don't want.