I think I figured out why I wasn't exactly great at Puzzle League. This game seems to favor more on players who are able to stay zen and intensely concentrate without getting distracted by anything, so they can consistently quickly spot for potential combos and chains and are able to plan ahead with much more ease.
That isn't something I'm particularly good at, and pretty much every single time I failed resulted from either panicking or having some attention deficit kicking in. At least, I reached near the end of story mode at Normal difficulty, completed Score Attack and Time Attack, and almost completed Candy Attack.
You could consider turning the volume completely off, as it did helped me quite a bit. The music that plays when the blocks are almost hitting on the top of the screen is so stressful it throws off my ability to concentrate, thus dooming my game. The thing is, getting a huge stack of 1x1 blocks feels advantageous because you've got more options on how you score combos and chains, and that particular music shouldn't be played at all.