Yes, I have a fix I use all the time. I use a piece of braided rope, 3/8th inch in diameter. If I fold over the end by about an inch and whip it with a needle and fine string, I can usually get it to pop under the gap and up the slot in the "round" version buckets and friction is enough to keep it there.
The "slot" buckets either need a slot cut up with side-cutters, or the rope to be fed through the slot, brought over the plastic, and sewn together.
Yes, having some sort of flat area in the middle is important for you hand. The firmer the better. I upcycle the round piece from the older buckets until they die, but very few new buckets have them as a separate piece instead of the formed type in your photo.
I use an old soldering iron as a hot knife to cut the rope into pieces that are long enough to hang down nicely, but not so long that the bucket keeps hitting the ground as I walk. I cut a bunch of pieces all at once, when I have a length I like.
Definitely worth the time to fix them in my opinion.