Sounds like "The Good Work", a short story by Theodore L. Thomas; first published in If, February 1959, which is available at the Internet Archive; also the (unaccepted) answer to the old question Utopia where no one has to work, but people have to work anyway. (It's really more of a dystopia.) Does any of these covers look familiar?
The story is much as you describe, except that the trivial tasks are tightening and loosening nuts.
"Yes, Mr. Winthrop. We have a job for you, and the full six hours a day, too. You will be on the maintenance crew of your building. Your job is explained here—" he passed over a card—"and it consists of tightening the nuts on the expansion joints in the framework of the building. It is very important to do it right, so read the card carefully." Winthrop nodded eagerly.
[. . . .]
Winthrop walked in and the two men stood looking at each other. Winthrop was surprised at how well Barlow looked, and he said so.
Barlow laughed. "Yes, the last time we met I was pretty far down in the dumps. But I'm working, Jeremiah. I'm actually working. Important work, too!"
His enthusiasm was infectious and Winthrop found himself laughing. "I'm glad for you, John. And I know how you feel, because I'm working too."
[. . . .]
There was an exchange of goodbyes, and Winthrop left.
Barlow went into the other room and came out immediately with his wrench. He waved it playfully at his wife. "Got to go," he said. "The loosening crew won't wait." And he blew a kiss at his wife and went off to work.