My gut reaction says no, mostly because it would make it very hard to attack at the same time, and because it would make the mount a more attractive target than the PC, which I feel is the opposite of the intended effect.
The actions are there to keep the mount alive, because it's a punch in the gut to kill someones' horse. The loss of gold resources alone is a bitter pill to swallow, but the loss of character concept and a (possibly) major NPC are even more potentially devastating. The rider benefits from the actions by basically exploiting forced movement and the clause that states opportunity attacks only happen if you use your movement. (Which I am fine with, otherwise mounted combat would be stupid)
However, it does make logical sense, and it would make mounted combat an extremely strong, if brutal to animals, option. You should play test them both a bit to see which one fits more with the themes of the campaign you are trying to run.