It's a matter of size and strength. Reindeer are small compared to horses - females average about 100kg, and males about 170kg. Horses used for riding are often 400kg and up. Some smaller horses are ridden, such as Mongolian horses/ponies, which are usually about 230-270kg, but these are still large compared to reindeer.

To illustrate the petiteness of reindeer:

• Dukha/Tsataan children riding: https://www.nomadasaurus.com/wp-content/uploads/2015/09/Little-Girl-Reindeer-1.jpg and https://www.nomadasaurus.com/wp-content/uploads/2015/09/Two-Boys-Reindeer.jpg

• Western adult riding reindeer: https://www.nomadasaurus.com/wp-content/uploads/2015/09/Jazza-Riding-Reindeer-1.jpg

• Adults on horses: https://www.nomadasaurus.com/wp-content/uploads/2015/09/Ride-To-Tsaatan-People-Old-Cam-2.jpg

(photos are from here, which has more cute reindeer photos and photos of the Tsataan people)

When the Tsataan/Dukha use reindeer as pack animals, they usually load them with about 30-40kg. The maximum loads are about 60-70kg. Only the larger reindeer are ridden by adults, and the added burden of 20kg of armour, helmet, shield, and weapons would be a problem. Thus, no reindeer cavalry.