Clement Clarke Moore, who wrote the poem originally titled A Visit from St. Nicholas & now generally referred to as The Night Before Christmas, was a dour straight~laced professor of classics who notably wrote A Compendious Lexicon of the Hebrew Language & refused to publish his poem as it was a mere trifle. He is remembered for his mere trifle. No~one remembers, let alone reads, his 2 tome academic achievement. It is in Clarke's poem that Santa's reindeer are first named. Rudolph is a later ring~in.
I know you've seen the cutesy pictures but reindeer are similar to caribou ~ a good sized, hefty deer that can stand as much as 4' high at the shoulders, weigh 250 pounds, travel at 50mph & pull twice its own weight ~ perfect for lugging sleighs round the sky all night! Reindeer are Santa's because they are the deer of the Artic Circle. There are no wild reindeer left, unless there is a remnant in far northern Norway. They have all been domesticated.
I was also fascinated to learn that the chances are very good that all Santa's reindeer are female ~ despite all those strong, male sounding names. Male reindeer shed their antlers before winter but females retain theirs until the spring. OK, some neutered & young bucks may also keep their antlers but they wouldn't be the biggest, strongest deer in the herd either.