This was datamined in 2015/2016. I’m surprised it hasn’t made an appearance yet

I think some attempts at racing were even vaguely prototyped during the initial beta or shortly after. But nothing ever came of it because, essentially, the riding skill stamina and speed mechanics determine absolute mount speed and sprint duration. Two characters with identical riding skill distributions will cover level ground at exactly the same speed. There's very little player-based skill that can affect it short of running into solid obstacles. So the only way to make it "competitive" would be to race on a very narrow path with only one ideal track, and anywhere else would be significantly more uneven (causing the player to ride greater distance due to large elevation changes) or outright walls to block progress. Imagine a slalom through a very narrow corridor with nothing but obstacles on either side, so that players have to jockey for position on the winning path.

In simple terms, mount racing would be like slot-car racing with perfectly matched cars. The winner is always the one that covers the shortest distance. So the only way to introduce variables is with obstacles.

The mount stamina changes on the PTS might encourage some minimal amount of player skill in managing sprinting, but it's at the cost of making the mechanic more annoying in regular gameplay.

Also, for any form of online game racing -- there has to be a way to deal with lag and the potential for lag-switching style exploits. Since ESO isn't fundamentally a racing game, the demands of engineering for that could be part of what makes it an unlikely addition.

Mounted combat -- especially in Cyrodiil -- could be awesome. But it would require a whole lot of additions to the combat system to determine how things work mounted vs dismounted. That would be a major change to the game code for combat, so it's a BIG deal to consider implementing.

Maybe ESO jousting could work. Two players ride toward each other on parallel tracks to take whacks at each other -- which causes damage/decreases riding stamina -- until the loser is knocked off their mount and has their health depleted. That might not require as much change to the way the game works; "mounted" combat would only occur in a specific context, and the existing stat system could be leveraged to determine outcomes and scoring. Sounds simpler on the surface, although it could still be difficult depending on the way the game must be programmed.

A steeplechase-style obstacle course that you'd perform while mounted would be awesome. There should be a reward of a mount for anyone who completes it.

Yeah, apparently the developers had some plans to do this at some point years ago, but never did.