It probably doesn't break the game.
Ready actions that fail aren't (or shouldn't be) so enormously common that this is going to seriously change the game. At worst, this makes readied actions slightly more attractive when the situation is fluid and it's hard to determine what exactly you might want to react to.
That said, it's not a huge thing, but I don't like this rule, personally -- it feels like you're sucking the wind out of the whole concept of a readied action, reducing it to merely "I feel like taking my action later on" rather than the player proactively setting up a cunning plan.
In my experience, losing Ready actions is a non-issue.
Usually, either the readied action isn't relevant or you can easily handle it on the fly without going so far as a full house-rule.
Consider a few common scenarios:
1) The trigger is directly associated with what you want to do in a way that doesn't allow for that sort of disconnect; if the trigger doesn't happen, then the action isn't relevant.
"I ready an action to shoot the monster when it pops out of the ground." (or "comes out of cover", "becomes visible", "enters melee range", etc.)
If the monster doesn't pop out of the ground, or if it surfaces somewhere where you don't have a shot, there really isn't a problem that your rule would solve. The exact trigger is irrelevant, because there's nothing to shoot anyway.
Could the monster pop out behind total cover and then move on the surface to come attack you, thus avoiding your trigger by not being a valid target when surfacing? Yes, that could happen, but that feels like the DM is just intentionally messing up the ready action. If it just happens to go that way and it wasn't intentional, then the DM can pretty easily assume that "when it pops out of the ground" really meant "when I have a clear shot at it" and allow the attack to proceed. Which is the next scenario:
2) The stated trigger was overly specific for what the player is clearly intending to accomplish.
"I put my sword to the viscount's neck and tell the guards, 'Back off if you want your boss to live!' I ready an action to cut his throat if any of the guards draw their swords!"
And then one of the guards pulls a crossbow on you and tries to shoot you in the face, and another one pulls out an axe, which is not a sword, and--
Look, this is the perfect time for the DM to just say, "Okay, yes, you meant if they grab a weapon, not swords in specific. Do you want to go through with it and cut the viscount's throat?" There's no need to house-rule in order to follow the intent of the trigger rather than the letter.
3) The player is surprised when things don't play out the way they expected, and the fact that the character doesn't get to do their thing reflects that they were unprepared for this situation.
"I ready an action to hit the first guy that comes through the door."
This reminds me of one of my favorite anime films, Lupin the 3rd: The Castle of Cagliostro. There's a scene where the protagonists realize they're about to be ambushed and get on either side of the door with weapons, ready to clobber the first guy that comes through -- and then a ninja crashes through the skylight to attack from the other side, putting them on the defensive, and only when they're tangled up fighting that one does the door open to reveal more ninjas.
Did Lupin and Jigen lose their readied attacks? Yeah, they did! They were surprised and it messed up their ambush, and that's how it should be. It would be absurd for them to just spin around and clobber the new threat they weren't expecting.
I do understand the issue here. If the player had been less descriptive with their wording and said something like "hit the first guy I can see" or "hit the first enemy who enters melee range", then it would be hard to deny them the attack against this unexpected threat. That does seem like it tends to discourage flavorful descriptions of ready actions -- but I don't think the house rule actually helps since it does away with the description entirely, in favor of just "I ready a melee attack". A possible solution is if a player loses their ready action because they just described their situation too well, the DM could give them Inspiration -- they improved to the game to their own detriment, and that's often what Inspiration is good for.
Well, okay, I guess there's a fourth scenario:
4) The DM is metagaming to intentionally screw over the player.
"No, no, you said 'swords', specifically, and nobody drew a sword!"
I mean, yes, this is a problem, but it isn't one you can fix with a house rule, because the DM is the problem.
House-rules or Behavior?
So, am I suggesting using your house-rule, just without telling the players about it? Well... sort of, but not really. To clarify my thinking:
Interpreting your players' actions leniently on a case-by-case basis is just part of good DMing -- and I mean that in general, not only in reference to Ready -- so you don't need to construct a specific rules backing to do that. To me, implementing a house rule is a larger and more serious step, and in this case, it doesn't seem warranted. What you want to accomplish here really boils down to simply treating your players with respect and maintaining the fun at the table. A house-rule implies there's a mechanical issue in the system that is serious enough and comes up often enough that it needs to addressed loud and clear to the players before the game even starts, but this doesn't seem to be that kind of situation.
I also see a pretty big difference between lenient interpretation of a trigger and just take the action any time you want. If a bulette un-burrows behind some rocks and comes running out into view, and you decide to allow the archer to take the shot when it enters view rather than sticking to the letter of "when comes out of the ground", that's not really the same as letting the archer just hold his shot until literally any time in the next round. They can't just stand there and wait to see how the Fighter's turn plays out before they decide to take the shot. One is the DM making a judgement call; the other is changing the entire nature of Ready actions.