The list of restrictions on the hound is extensive and excludes much of what mounts are useful for:
The hound appears in an unoccupied space of his choice within 30 feet of the target. Launch initiative for the hound. In his turn, he can move only towards his target by the most direct route, and he can use his action only to attack his target. The hound can make attacks of opportunity, but only against its target. Also, while the hound is less than 5 feet from the target, the target has a disadvantage when saving casts against any spell you cast. The hound disappears if it is reduced to 0 hit points, if its target is reduced to 0 hit points or after 5 minutes.
- you can only move towards your goal by the most direct route
- You can only use your action to attack your target, and you can only make attacks of opportunity against your target
- only lasts a maximum of 5 minutes
So, to answer your question, no, it does not move as it directs it, it moves in a straight line, through objects and creatures if necessary, towards its objective and can only use its action to attack its objective.
A slightly more liberal DM may also allow him to launch himself at his target if he is too far away to attack him … but that depends on the DM.