The direct, strict reading of the rule says no: but this DM thinks that is an unrealistic rule.

As noted in the question, a controlled mount can take only 3 actions: Dash, Disengage and Dodge, but ending a grapple, among other possibilities, requires spending its action to make a DEX or a STR check. Unluckily, this is not permitted by the mounted combat rules.

The argument specific beats general seems to suggest that the grappled condition is more specific than the mounted rules: at a first look, it seems reasonable. Indeed, the grapple condition allows to spend own action for trying to break the grapple, it might seem to add a further option to the list of the allowed actions.

Anyway, this is not true: it merely describes how you can break the grapple, but this requires to spend the action in a particular way (STR or DEX check), which is not allowed per the mounted combat rules, the limitations are not overridden by the grappled condition. It sounds pretty strange that having a disadvantageous condition (grappled) enlarge the number of action options.

How this DM shall rule in this case.

It seems really strange to me why the designers decided to write the mounted combat rules in this way, maybe there are some shenanigans that I am not aware of.

I would allow the player to command the mount to try to break the grapple, disregarding the actual rules, since in this case it seems to me more than reasonable. Moreover, I would allow even to make a free WIS (Animal Handling) check: if the result is high enough (for example, 5 or more higher than the grapple DC), I would allow the mount to make the DEX or STR check with advantage. Or maybe the player may just use its action to take the Help action.

Of DMs and Rules.

This answer from Jack touches an important fact of Dungeons and Dragons. The game is made of written rules: some of them are really reasonable, others not so much, others are pretty strange. The role of the DM is to adjudicate when the rules make sense for the table and for the players, and how to eventually modify them, paying attention to create a balanced game and not to favor too much the players (the game becomes boring) or to render it too difficult (it becomes frustrating).

Here, I truly believe that we are in the case of "strange" rules that needs DM intervention.

It can certainly try!

It's the DM's job to adjudicate how.

Frame Challenge

A bit of a frame challenge, though.

Your question implies that only what is specifically described in the rules are the only things that can happen. That's not what the rules say.

From the DMG introduction, under "How to Use These Rules":

The rules don’t account for every possible situation that might arise during a typical D&D session. For example, a player might want his or her character to hurl a brazier full of hot coals into a monster’s face. How you determine the outcome of this action is up to you.

Your question implies this:

DM: Your mount has been grappled. Unfortunately, the rules do not account for a mount being grappled. You're stuck.

The introduction to the Player's Handbook describes "How to Play":

1. The DM describes the environment.
2. The players describe what they want to do.
3. The DM narrates the results of the adventurers’ actions.

DM: Your mount has been grappled. Player: Uh oh. Can it break free?

It is now the DM's job to determine whether it's possible for the mount to break free.

But I want to play RAW

Clearly you're concerned that a controlled mount can only take three actions: Dash, Disengage, and Dodge.

Understandable. Don't get fixated on that one rule compared to the rest of the rules, though.

The PHB explains this under "Actions in Combat":

When you take your action on your turn, you can take one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise. Many monsters have action options of their own in their stat blocks.

When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.

Guidance

If you're a player

If you're a player, and you're concerned about your mount getting stuck, talk to your DM.

If you're a DM

You might be wondering how to adjudicate the horse getting out of the grapple.

That is easy!

The horse is grappled by . . . something, but it is not the rules.

Using the grapple rules to allow the horse a chance to escape is perfectly reasonable. You're not breaking RAW, or changing the rules, or anything.

Or, if you don't like the grapple rules, if for some reason you don't think they're applicable here, it's reasonable to use some other test.

The grapple rules have the great advantage of being applicable in many situations.

If you're neither a player or a DM

If you're reading the rules hypothetically, well, you can't play without a DM, you need to fulfill the role of the DM yourself.

But what about the limitations of mounted combat?

If you want to interpret "only three action options: Dash, Disengage, and Dodge" as trapping the horse, go for it. But they don't. The rules do not provide for a grappled, controlled mount.

The rules are wrong!

You've certainly found a gap.

I go back, over and over, to what I quoted at the beginning:

The rules don’t account for every possible situation that might arise during a typical D&D session.

The rules for mounted combat are pretty simple. They just don't account for a controlled mount to be grappled.

If you're a rules lawyer

I get it.

Some answers say, "the direct, strict reading of the rule says no". I disagree, I think the rules don't say. The rules don't cover a great many situations.

Some answers say, look at it this way, look at it that way.

But it doesn't really matter. It seems like most or all of the answers come to the same conclusion:

Yes, the mount can attempt to break the grapple, and the grapple rules are a good way to do it.

Only if the rider chooses to allow it to act independently.

While you’re mounted, you have two options. You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.

Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes.

emphasis mine

Of course, if you allow it to act independently, your DM may choose to have the mount do something else entirely. Arguably, a grappled mount may no longer be "controlled" in the sense that movement is now 0, precluding controlled actions. Ask your DM.

Strictly RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

Because Dash, Disengage, and Dodge are the only actions a controlled mount can take, it cannot use its action to escape the grapple while it is controlled.

This is the strict reading of the rules, but your DM might rule it differently.

Side note

If the rider doesn't control the mount, it acts independently and is no longer limited to specific actions. In that case, it could use its action to attempt to escape a grapple.

Addendum: Why this is not a Specific vs General conflict

The specific vs general principle applies when two rules attempt to answer the same question in different ways. Here, they address different things.

• Grapple rule (general): A grappled creature can use its action to escape. This rule permits an action; it does not guarantee that the creature always has access to that action in every circumstance (i.e. a creature that is incapacitated or stunned can't escape a grapple even though the grapple rules say it can use its action to do so)

• Controlled mount rule (specific to controlled mounts): A controlled mount can take only Dash, Disengage, or Dodge. This rule restricts what actions are available in a very specific state: being a controlled mount.

There is no conflict between these rules. As a result, this is not a specific-vs-general issue. The grapple rules define the cost of escaping, while the controlled mount rules limit which actions are available. Since Escape is not an available action for a controlled mount, there is nothing for specific-over-general to resolve.

Yes.

Specific beats general.

This book contains rules [...] that govern how the game plays. [...] If a specific rule contradicts a general rule, the specific rule wins.

The Mounted Combat rules describe a general situation—how movement and positioning work when one creature rides another.

By contrast, the Grappled condition is a defined mechanical state with explicit rules that apply only while that condition exists. Conditions are designed to override normal circumstances.

The Grappled condition explicitly grants a mechanical permission:

Escaping a Grapple. A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

Since mounted combat does not specifically remove this permission, a grappled controlled mount may use its action to attempt to escape.