Let's take a look at the text:
The initiative of a controlled mount changes to match yours when you
mount it. It moves as you direct it, and it has only three action
options: Dash, Disengage, and Dodge. A controlled mount can move and
act even on the turn that you mount it.
Okay, first off, "the initiative of the controlled mount changes to match yours." This makes sense, since it will "move as you direct it," which implies two things: first, that it will (normally) only move as you direct it and will not make decisions on its own, and, second, it uses its movement as movement, not an action. Some of the other answers here seemed to suggest that controlling the mount would take an action, but if that were the case, how does the Mounted Combatant feat make any sense? Specifically, how would you be getting Advantage on Melee attacks while riding your mount if you are using your action to direct the mount instead of attacking?
Okay, moving on. The next thing it says is that your mount "has only three action options: Dash, Disengage, and Dodge." This specifically states that the MOUNT still gets an action for the round, an action that you direct it to take, per the previous sentence, and that uses the mount's action, not yours, however, you are not permitted to use the mount's action to attack.
Finally, the last sentence states that your "mount can move and act even on the turn that you mount it," which further supports the idea that as the rider, you direct the mount's movement and action each round independent from your own action. Your own action can then be used to attack.
Personally, I would rule that the three listed actions you can make your mount perform in combat are listed as freebies, that is, they are the only ones that can be preformed using the mount's action, and to make your mount do anything else, counts as a "risky maneuver". In the Using Skills section in the PHB (p.178) it states that "You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver." Since in most cases, performing a Skill check uses your action for the round, I believe that it is reasonable to allow a character who is riding a trained mount to attempt to urge their mount to attack by using their action for the round to attempt a Wisdom (Animal Handling) check. I would make the DC for the check be equal to the AC of the target being attacked and give Disadvantage if your mount is not combat trained.
I feel like this ruling provides the best answer that allows the DM to say "yes" to the player without breaking the mechanics of the game. It adds utility to the Animal Handling skill, and rewards players for investing in the skill. It allows players with low Animal Handling skill to still attempt using a combat mount, just as a wizard can still try to attack with a great sword; it probably won't work out too well for them, but they can try.