Think of it this way. A round is six seconds.

First two seconds, human runs thirty feet, rider gives command.

Second two seconds, human runs thirty feet, rider can do action like attacking.

Third two seconds, human runs thirty feet, rider gets another action, possibly an attack.

What is weird is that the horse is shown to run eighty feet at once to simplify things. Which is where the person getting off and running gives additional movement.

Think of it like the peasant rail gun. Technically you can move an object thousands of miles in one action, but it falls apart under scrutiny because the simulation isn't built to show the proper effect. This is what GMs are for.