I use 2 1/2" binding and never use a walking foot -- no puckering issues with binding.

I believe this has to do with the stretch of the binding fabric, not the width of it.

I looked really close at your problem, because I had never had it. My only suggestion is to use bias binding instead of cutting it straight. There are several ways to cut bias binding, but my favorite is the one where you cut a yard of fabric and even the edges, fold the piece on the diagonal, then using the fold as the top edge, fold the sides into the middle, and fold the whole thing in half again. Your folded fabric should now be narrow enough to lay a ruler along the top edge and trim 1/4” from the edge. Now continue cutting 2 1/2 “ strips down the entire length. You will have enough strips to do a queen quilt, and extra to save for your next scrappy quilt.
The great thing about bias binding is no scootching and puckering. I have done enough sewing through the years to recognize that trying to use a straight binding would just frustrate me the way yours does you. Good luck.

I sew my binding ends so they're snug and then iron down the unsewn section of binding with a few dots of Elmer's glue, it keeps it perfectly in place and no bunching.

I had that problem, too. I tried this new to me method and the bunching problem was solved.
At about the 4:30 mark she describes how to do the final join the in the binding. I use a generous 1/4 inch space and it works every time .

https://suzyquilts.com/how-to-sew-binding-on-a-quilt/