I assume it’s that leg healing reduces their speed or endurance

Basically this, part of how they get their speed (legs being mostly tendon and cartilage) is what makes healing so difficult. Both tendons and cartilage have far less blood supply compared to muscle or skin, which heal much more quickly. So if a horse developed legs with more tissue and muscle (and thus an increased blood supply, improving healing), they would have been slower than their skinnier legged cousins and had no competitive advantage.