He rewards good deeds and through history punished bad deeds with a beating from a krampus (which became elves later)
Rewarding good deeds isn't a bad thing, and I don't know anyone who tells their kids to be good or else they'll get beaten by elves. I've never even heard of a krampus.
The Krampus is a German thing. No one without German roots has it as part of their tradition. On the other hand, the theme of mischievous beings out and about around this time of year seems quite widespread, like the kallikantzaroi
of Greek folklore.
Yesh is (again) generalising something local. It is traditional in most cultures to exchange presents around the change of the year, but who brings them and exactly when varies, a lot. It would be a pity if the American idea (as opposed to just the imagery) of Santa supplanted everything local.
Krampus turned into "blackjacks" and later made into Elves by Coca Cola.