Brother RP (and to a far, far lesser extent, Didymus, I guess): Thank you for your support! As usual, GOD gives us all the answers if only we search hard enough and open our hearts to Him.
Mrs. Katydid: Where in the Bible does it say that thumbsucking IS permitted? (Answer: nowhere!)
As for teddy bears and security blankets ... well, GOD is pretty clear about how He feels when people worship anything other than Him.
(Leviticus 26:1; Exodus 20:5)
So basically, children who rely on stuffed animals and ratty shreds of fabric to soothe them (instead of relying on the LORD), are not only damning themselves but also four generations of brats that come after them.
So the "problem with it" is the child's eternal soul ... and their children's ... and their children's children ... you get the point. There is nothing "unreasonable" about wanting your children to be Saved and rejoicing with Jesus in Heaven, rather than being poked by Satan down below. The answer here is to remove the child's thumb(s) if they won't stop the sucking.
Anyone who feels otherwise is clearly a child-hating, God-flauting sadistic heathen.