I'm trying to design a system for horse-riding an mounted combat for a 2D platform game, and I'm kind of stuck. The flat nature of the game space makes it difficult to give the combat sufficient tactical depth, and I don't want to end up with a glorified quick time event. Can anybody think of any good instances of riding and ideally mounted combat in a 2D platformer or related genre?

How much of the game depends on horse-mounted combat? If is just a small part, I would not overthink it.

If it is a significant part in duels, take a look at jousting, having both horsemen with a shield and a lance. The player would balance defense (position of the shield) and offense (his lance position), and the horse speed. Think For Honour stances. Check Nidhogg to have an idea on how this could work on 2D. It should be possible to extend this system to work on battles (cavalry charge), adding sword combat for close range, or when the player looses the lance.

Combat on horseback should be physically very similar to combat for characters on foot: horses are taller, faster, harder to steer, but walking, running and jumping like people.

Unless you plan to feature activities that might warrant a special user interface or subgame (jousting, circus-style acrobatics, dressage, horse racing etc.) or special obstacles and situations for which a horse is required (e.g. jumping over a moat) you should just design game rules and environments that work for infantry and cavalry in similar ways.