Why wasn't Rudolph included with Santa's reindeer?

Why wasn't Rudolph included with Santa's reindeer?

There are two reasons, and they go hand-in-hand.

Reason #1

Just as you pointed out, Rudolph isn't a member of Santa's original reindeer. The idea of Santa having reindeer to usher him around the world was first introduced in 1822, and then, in 1823, the poem "Twas the Night Before Christmas" mentions them.

... More rapid than eagles his coursers they came, And he whistled, and shouted, and call'd them by name: "Now, Dasher! Now, Dancer! Now, Prancer, and Vixen! "On, Comet! On, Cupid! On, Dunder and Blixem!

"To the top of the porch! to the top of the wall! "Now dash away! dash away! dash away all!" ...

Rudolph, however, wasn't conceived of for more than 100 years later (1939). Given this fact, it could be argued that, if someone is wanting to tell a story that includes traditional or classic Santa, they may feel inclined to exclude Rudolph.

Reason #2

When Scott is reading to Charlie while in bed (on Christmas Eve), if you notice, the book that Scott's reading is in fact the story "Night Before Christmas". At one point, Charlie asks what "a rose suchak ladder" means, but Scott corrects him to say "arose such a clatter". Shortly after this though, Charlie and Scott inspect a ladder outside their house, with the engraving "a rose suchak ladder" on it. This then suggests that some parts of the nighttime story are actually coming true, and this would also necessitate the exclusion of Rudolph.

To summarize..

If someone is wanting to tell a story that involves traditional Santa, Rudolph probably wouldn't be included. Additionally, given that the film specifically uses "The Night Before Christmas" as a platform for transitioning into a kind of fantasyland, it makes sense to honor and stay true to a critical aspect of that story -- Santa's reindeer.

Rudolph the Red Nosed Reindeer was originally a marketing gimmick, used by the Montgomery Ward Department Store. It was conceived by Robert May to be used as a character in a coloring book given away by the department store at Christmas time, since it was cheaper to create your own characters than to license established ones.

The story, and character, is currently owned by The Rudolph Company, LP.

Since the movie included Scott (Tim Allen) reading "The Night Before Christmas" to his son Charlie (Eric Lloyd), and the story was written prior to Rudolph's creation, Rudolph was not part of the story. Thus, it made no sense to include him in the movie at an additional expense (since it would require a separate licensing agreement).