"Q. U. R.", a short story by Anthony Boucher which was also the answer to the old question 1970's or earlier SF short story - unique cocktail drink that robots can't duplicate. Originally published (as by H. H. Holmes) in Astounding Science-Fiction, March 1943, available at the Internet Archive. You may have read it in one of these compilations. There was a sequel called "Robinc".
Reminiscence:
It's got so the young sprouts nowadays seem never to have heard of androids. Oh, they look at them in museums and they read the references to them in the literature of the time, but they never seem to realize how essential a part of life androids once were, how our whole civilization, in fact, depended on them. And when you say you got your start in life as trouble shooter for an android factory, they look at you as though you'd worked in twodimensional shows way back before the sollies, as though you ought to be in a museum yourself.
The problem:
Because my job wasn't one that you could carry on comfortably in conditioned buildings and streets, it meant going outside and top side and everywhere that a robot might work. We called the androids robots then. We hadn't conceived of any kind of robot that wasn,t an android or at least a naturoid of some sort.
And these breakdowns were striking everywhere, hitting robots in every line of activity. Even the Martoids arid Veneroids that some ex-colonists fancied for servants. It would be an arm that went limp or a leg that crumpled up or a tentacle that collapsed. Sometimes mental trouble, too, slight indications of a tendency toward insubordination, even a sort of mania that wasn't supposed to be in their make-up. And the thing kept spreading and getting worse. Any manifestation like this among living beings, and you'd think of an epidemic. But what germ could attack tempered duralite?
The solution:
"Why, isn't it obvious?" he asked simply. "When Zwergenhaus invented the first robot, he wasn't thinking functionally. He was trying to make a mechanical man. He did, and he made a good job of it. But that's silly. Man isn't a functionally useful animal. There's very little he can do himself. What's made him top dog is that he can invent and use tools to do what needs doing. But why make his mechanical servants as helplessly constructed as he is?
"Almost every robot, except perhaps a few like farmhands, does only one or two things and does those things constantly. All right. Shape them so that they can best do just those things, with no parts left over. Give them a brain, eyes and ears to receive commands, and whatever organs they need for their work.
"Q.U.R." stands for "Quinby's Usuform Robots". In this passage they talk about designing a robot bartender:
We watched entranced as he mixed the potion. "Get exactly what he does," Quinby had said. "Then construct a usuform bartender who'll be infallible. It'll satisfy the Martian envoy and at the same time remind the Head of why we're helping him out."
