A saying similar to "playing whack-a-mole"

"What is important is seldom urgent and what is urgent is seldom important."

Dwight D. Eisenhower

But your literal translation does sound better.

putting out fires

Definitions with example sentences:

put out fires (Cambridge Dictionary)

US

(UK firefight)

to spend time on problems that need to be dealt with quickly, instead of working in a calm, planned way:

• She spends much of her time putting out fires and navigating red tape.

See comment below re: firefight ("firefighting mode").

put out fires (Farlex Dictionary of Idioms)

To deal with emergencies or urgent matters rather than ordinary day-to-day tasks.

I spent so much time today putting out fires with our suppliers that I didn't even have a chance to read my emails.

Another take on putting out fires (VOA, Grant Barrett):

If you spend your days putting out fires, and you mean this in a metaphorical way, it means that you've got a lot of small problems that you have to deal with all together. And so you spend a lot of time just kind of solving problems one after the other, without a lot of chance to kind of catch your breath and think about the larger picture of what you're doing.

Quoted per broadcast (WORDMASTER), not article with typo.

The Spanish expression is at least partly wry, as urgent matters need immediate attention. But there must be an interpretation of 'urgent' as 'usually seen as number one priorities, and quite possibly wrongly' here. Thus:

The good is the enemy of the best

Settling for things that are merely good or adequate can prohibit one from achieving that which is ideal.

• My opponent wants our government to provide only enough funding for "good" services—good schools, good hospitals, good infrastructure, and so on. But good is the enemy of the best, and that is why he and his party will never lead our country to its fullest potential.
• You should never allow your employees to get away with mediocrity, for the good is the enemy of the best, and if left unchecked it will hinder your company's ability to grow.

[Farlex Dictionary of Idioms. © 2015 Farlex]

It equates to wrongly (but understandably so) prioritising.

When you're up to your neck in alligators, it's hard to remember you came to drain the swamp might work.

It is easy to lose sight of one's initial objective, becoming caught up in subtasks or in tasks only tangentially related to the original goal.
[Wikitionary]

@EdwinAshworth's comment about the poem "Leisure" by W. H. Davies made me remember this quote from John Lennon:

Life is what happens to you while you're busy making other plans.

The quote is taken from the lyrics of "Beautiful Boy".

There's a couple common phrases that might apply to concentrating too much on the small details and ignoring the more important bigger picture:

Penny wise, pound foolish

(Wikitionary)

(idiomatic) Prudent and thrifty with small amounts of money, but wasteful with large amounts.

(A little UK-centric, but common enough phrase to be understood in the US and elsewhere.)

Can't see the forest for the trees

(Wikitionary)

(idiomatic) To discern an overall pattern from a mass of detail; to see the big picture, or the broader, more general situation.

(idiomatic, in the negative, by extension) To be overwhelmed by detail to the point where it obscures the overall situation.

Smith is good at detail, but can't see the forest for the trees.

This is almost always used in negative constructions, often starting with can't, as it is a negative polarity item.

The thing that came to mind for me was the "tyranny of the urgent." Looking it up, it looks like there's a long quote by Charles Hummel that gets across pretty much the exact same idea, even using the same words "important" and "urgent":

The important task rarely must be done today, or even this week...But the urgent task calls for instant action...The momentary appeal of these tasks seems irresistible and important, and they devour our energy. But in the light of time's perspective, their deceptive prominence fades; with a sense of loss we recall the vital tasks we pushed aside. We realize we've become slaves to the tyranny of the urgent.

Source: https://www.azquotes.com/quote/1062500

An expression became popular a number of years ago, and it was

The tyranny of the urgent

The expression means that there are times when the lesser important things in life become so demanding that we let the truly important things slip by us. Some relevant examples of things that tyrannize us in 2021 are emails, particularly those that lure us into opening them because there is something free involved; voicemails; telemarketing calls; Facebook updates and postings of friends; and so much more.

All of the above may seem urgent at the time we become aware of them, but in terms of what is really important to us, such as relationships with family and friends, they are of little importance. They are also time wasters and energy sappers.

So if your wife, Cascabel, wants to use the phrase in everyday conversation, she might try saying, for example,

Sounds to me, my friend, you are being tyrannized by the urgent!

A related saying is

festina lente (latin)

or

make haste, but slowly

Wikipedia has a nice article on this https://en.wikipedia.org/wiki/Festina_lente

The meaning of the phrase is that activities should be performed with a proper balance of urgency and diligence.

Sometimes you will do well just to translate the expression directly into English, and I think this is one of those times. The meaning is clear from the words themselves, and it will stand out precisely because it is not a stock phrase in English.

I will say that the concept that urgency and importance need to be considered separately has become a commonplace in management talk. But the expression "The urgent does not leave time for the important" (or perhaps "is not leaving time") will have some freshness, I think.

What I've heard that seems appropriate is:

There is never enough time to do it right, but there is always enough time to do it over.

I'll admit the literal translation was good sentence even though it might not be a familiar saying in English.

In American English, the phrase "In the weeds" refers to being unable to catch up on your tasks because more small tasks keep being added to the list. In particular, this is common in restaurant work, where a server may have to refill water glasses, deliver food, take orders, drop checks, and bus dirty dishes on many tables at once, and ends up providing poor service and earning lower tips as a result.

Although this is very common in food service, I have heard it in many other contexts where someone feels overwhelmed with minor tasks and is unable to focus on longer-term or more important large considerations. Personally, I picture walking through a field where the plants around me are literally so high I can't really see where I'm going, I can only see the weeds right in front of my face. I don't know if this is the real origin though.

Person 1: "Hey, should we refinance the house? I think we can get a good interest rate, and maybe we could start contributing more to retirement with the money we save!"

Person 2: "Oh, I am so in the weeds dealing with teaching my kids at home during COVID lockdown that I can't even think about that right now. Can we talk about it next month when I have more time to think?"

Your wife's saying speaks more to the tendency of immediate problems to take time and resources away from more strategic endeavors, and implies that all the while the anxiety of the important being neglected is acknowledged and felt. There is awareness of the problem.

I concur with her assessment of "Whack-A-Mole" in that it only addresses half of the issue; it's all urgency and no importance.

"Penny wise, pound foolish" (or, occasionally, "penny wise, dollar stupid" here in the colonies) is closer, but that one speaks more to the skewed intentions/perceptions—and thus the decisions and actions—of the person or entity to whom it is applied; it does not consider a situation wherein it is the circumstances themselves which lead a reasonable person to overcommit time and resources to the immediate at the expense of the important. The penny wise, pound foolish person is generally not aware of their state, and so feels no anxiety about their pound-foolishness.

I'd say you have the best one. If anything, perhaps shorten it even further, to:

"The urgent leaves no time for the important."

There is always the colourful expression When you are up to your ass in alligators it's difficult to remind yourself that your objective was to drain the swamp and its variations. This carries the same connotation of being too busy dealing with a host of urgent problems to get on with the main task.

It's fairly obviously American in origin (we don't get many wild alligators in the UK) but it has travelled quite well.

After I posted this I saw Decapitated Soul's similar answer from 4th Feb. See my comment on his answer for my reason for not deleting this one!

A 'Quick fix' solution comes to mind. There is anther one related to plasters/band-aids but I can't remember it.

Quick fix: : an expedient usually temporary or inadequate solution to a problem. www.merriam-webster.com › dictionary › quick fix

'When it comes to correcting a problem in your organization, you should make sure you are, in fact, fixing the problem and not just a symptom. In this article, Esther Derby takes a look at the issue of the quick fix... (https://www.agileconnection.com/article/fixing-quick-fix).'