Apparently, it was all about that particular rig/ging. I made another simple test in a new file, and voila...
I've tried modifying parenting, binding, and several different settings, but to no avail. And once I removed the Angle constraints for those hip bones, it all worked as expected.
My idea was to have just the legs affected by squash&stretch, so to preserve the character's design (a flat line at the waist and legs in the given positions), I had to add another bone, from the root bone. I've tried with just 2 bones per leg - 1 from the root, aiming for the knee and the 2nd one as a shin but to no avail again. Manual adjustments were also required.
Even if I remove the hip bones and bind the waist and top of the legs to the root bone - these additional bones (1 per leg) are still needed. The thing is, once the stretch begins, all of the proportions are gone and have to be adjusted manually. Since the SS is not actually necessary in this case and since I'd have to adjust the deformations manually - why bother adding Max IK stretch at all? What am I missing?
*** solution ***
Adding a pin bone between the hip bone (with constraints on) and the upper leg bone solves the issue.
* I believe it's not shown, but I've edited the message several times. Which, I believe, can be seen x)