I was inspired by reading the Sidekicks rules in Tasha's to attempt a homebrewed improvement to horses, to allow a PC who just wants to be a credible threat on horseback access to that, without putting the necessary hoops behind too many gates, so there's widespread access to it.  Feedback is appreciated.  I'll be using the warhorse as a base; things will definitely get weird if you try extending these rules to anything Int 4 or higher.  Here are the stock rules on mounted combat, for reference.

1. Horses are considered to be level 1 for purposes where this is necessary to know, such as calculating their proficiency bonus (2).
2. A rider proficient in Animal Handling can take the Help action to Help their mount on Athletics and Acrobatics checks.  Likewise, if the rider is proficient in Animal Handling, the mount can take the Attack action, even if it's acting on the rider's initiative, and the same applies to the Help action for helping an attack.  If the rider is proficient in Animal Handling, when they take the Ready action, their mount, when acting on the rider's initiative, can also take the Ready action - the mount's trigger is being nudged by the rider, and what the mount does is specified by the rider as normal.
3. A horse can be trained to be proficient in armor, so it can wear barding without suffering nonproficiency penalties.  This uses the Xanathar's training rules, except an int penalty does increase the time needed, so it takes 14 workweeks to train the horse, and it must learn in order: light->medium->heavy.  You can hire an NPC to do this, but they will charge 25gp per workweek to train it for you (350 gp total).  If a PC does this, they need proficiency in Animal Handling, and their Wisdom bonus reduces the total workweeks needed.
4. Quadruped: as a quadruped, a horse has Powerful Build, so it counts as one size category larger for calculating carrying capacity.  It is challenging to knock a quadruped prone, and they are naturally fast, but horses are not built well for climbing.  All horses are proficient in Acrobatics and Athletics, but they have disadvantage on checks made to climb, and climbing movement costs them quadruple normal, not twice normal (octuple normal in difficult terrain).  Horses don't have arms or hands; they can't reach above them at all, per the jumping rules, and the GM is encouraged to rule many surfaces a biped can climb are simply unclimbable - e.g. a horse cannot climb a vertical ladder.  When in doubt, assume a horse can't climb.
5. Horses are proficient in Strength and Constitution saves, and in Dexterity saves made to avoid being knocked prone, such as when traversing scattered ball bearings.
6. The above rules qualify a warhorse that's been trained in all 3 barding levels as a Warrior Sidekick, but Martial Role was replaced with additional save proficiencies.  Horses level from here in their new Sidekick Class, Horse.

Horse abilities by level:

1. See above.  The Tasha's rules are clear as mud on whether a horse should gain a hit die upon gaining its first Sidekick level, so feel free to add 1d10+1 hit points to the Warhorse if you interpret the rules as saying yes to this.
2. Animal Cunning: The horse can now Dash or Disengage as a bonus action, and a rider proficient in Animal Handling can Help the mount as a bonus action.
3. Expert Movement: The horse now doubles its proficiency bonus in Athletics and Acrobatics.
4. ASI:  A horse cannot take an ASI bonus to Intelligence or Charisma, and should not be able to take feats.
5. -
6. Agile:  The horse is now proficient in all Dexterity saves, and has advantage on Strength and Dexterity saves to resist being knocked prone.
7. Improved Instincts:  The horse is now proficient in Perception and Initiative checks, and can calculate its Initiative using its Wisdom or Dexterity modifier.  A mounted rider proficient in Animal Handling can use their mount's initiative roll instead of their own when initiative is rolled.
8. ASI.
9. -
10. Reliable Movement: The horse now treats Athletics and Acrobatics rolls of less than 10 as 10.
11. Loyal:  While ridden by a rider proficient in Animal Handling, the rider can use their reaction when the horse would make a Wisdom save to replace the save with the result of the rider's Animal Handling (Wisdom) check.
12. ASI.
13. -
14. ASI.
15. Improved Loyalty: When Loyal is applied, the rider can choose for the result to continue to apply to all of the horse's Wisdom saves until the horse's next turn, but doing so makes the horse suffer Disadvantage on Intelligence and Charisma saves for the same duration.
16. ASI.
17. -
18. Greater Loyalty: Extending Loyal's duration no longer applies Disadvantage on Intelligence and Charisma saves.
19. ASI.
20. A level 20 horse is immune to being frightened or charmed.