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According to the Player's Handbook, pg. 198:

Controlling a Mount

While you're mounted, you have two options. You can either control the mount or allow it to act independently.

You can control a mount only if it has been trained to accept a rider. [...] 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.

This does not specify if the actions listed:

1. are the only options a controlled mount may take in the round, OR


2. are the only options a controlled mount may take with, explicitly, a standard Action.

Can a controlled mount make an Opportunity Attack if an enemy moves away from it?

Yes, it can

The relevant rules for opportunity attacks (from the Basic Rules):

You can make an opportunity attack when a hostile creature that you
can see moves out of your reach. To make the opportunity attack, you
use your reaction to make one melee attack against the provoking
creature. The attack interrupts the provoking creature’s movement,
occurring right before the creature leaves your reach.

As you've quoted, controlling a mount does not place any restrictions on the mount's freedom to use its Reaction.

But as a DM, I probably wouldn't allow it

I interpret a controlled mount to be fully under the control of the controlling character. A mount that is free to react as it chooses is by definition not under control. If the rider wishes to instruct it how to react, that would use the rider's own reaction, and so the rider would have to forgo her own opportunity attack.

Yes, it can

The only limitations on a controlled mount are “... It moves as you direct it, and it has only three action options ...”. It’s bonus action and reaction (and legendary and lair actions) are not limited so it can use them to do whatever it could normally use them for, like taking opportunity attacks.