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

My wife is looking for a phrase or saying in English that is similar to...

Lo urgente no deja tiempo para lo importante

...which means "Urgent matters do not leave time for what is really important", or more simply,

"The urgent does not leave time for the important"

I suggested "playing whack-a-mole", but she says that doesn't express the idea of "not having time for what is really important".

Any ideas?

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.

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]

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

Dwight D. Eisenhower

But your literal translation does sound better.