Yes the writings of the Christin's write as if everything was their idea but there is sufficient evidence of persistent rituals and archeologic evidence to refute this.
Thus you look at things on a case by case basis.
Some local goddess reappearing as a local saint of the same name with the same story in a relatively short space of time is very different from some Germanic goddess reappearing as a Greek bishop and later saint who was venerated since before the Germans were even Christianised across multiple religious and cultural divides.
Gift giving dates much farther back than 400 years in cultural celebrations and the way it is portrayed to day is much more in keeping with Frau Holle (as well as different names in different regions) that with St Nicholas who was imported to America through the Dutch who would have been well known to the myths of Northern Europe which continued into the 1800's.
Not in Christmas celebrations it doesn't, which is the point. Gift giving can't be claimed as 'pagan', only an attempt to appropriate an existing feast day or figure.
Other than the Frau Holle contention being both tenuous and speculative, St Nick doesn't resemble an attempt to appropriate another figure in the manner you suggest.
1. St Nicholas has a long, unbroken record of being an important saint associated with acts of charity and children (he actually reanimated some
)
2. He is not a Northern European saint, but an Anatolian one that only reached Northern Europe late, via Italy. He was celebrated far outside the cultural reaches of Germanic paganism
3. Gift giving was associated with St Nicholas's day, 6 Dec, not Christmas
4. It moved to Christmas as venerating saints was "idolatrous Popery" and thus despised via the English figure of Father Christmas.
5. The hardcore Protestants who denigrated St Nicholas' day as "idolatrous Popery" would not have missed a chance to denigrate it as pagan had they had the remotest inkling it was pagan. They made up all kinds of pagan links after all, but not a peep on this one, despite being aware of myths involving Frau Holle.
6. The Dutch Sinterklaas didn't have reindeer
7. The reindeer and stuff were artistic representations in 19th C America and it is very much a stretch to assume these were 'pagan symbolism' without any evidence. Artistic licence can be just that: one person's imagination.