Specifying these elements precisely will reveal if they really are such opposed goals, and identify how they might be maximally satisfied
I would say that, at a glance, none of these goals are diametrically opposed in the manner the question suggests. For example, I see no reason that service can't be prompt and relaxed from the point of view of a customer. If we imagine four broad combinations for that situation:
- Long wait to be served + harried employee = impression of a really busy environment
- Long wait to be served + relaxed employee = impression of a lazy employee
- Quick service + harried employee = impression of a frantic environment which is running smoothly with a hard working employee
- Quick service + relaxed employee = impression of a competent employee and well managed store equipped to handle its workload
If you believe that #4 is legitimately impossible, then it's time to discuss which elements we're each defining as constituting quick service and relaxed environment respectively. Doing so will make any necessary tradeoffs between the two clearer.
I think that the key here is to understand what your clients functionally want, rather than literally interpreting their expressed desires. For another example, a client wanting to understand a system completely without any technical talk is probably using a different definition of "complete understanding" than you are.
I would argue that most people have a fairly complete understanding of their stoves with respect to their anticipated use-cases: they know how to achieve any effect they want (turn on any combination of burners, adjust heat, etc.), but may not be able to describe anything about how the stove actually operates, and might be completely hopeless at trying to repair it if it broke because they don't understand it to that degree. A layperson who wants to learn "everything about their stove" probably means "everything about the operation" of the stove, and may not appreciate or be interested in a technical discussion of what components make the stove work, and how.
If you're having these things asked of you, it is incumbent on you, yourself, to precisely identify what clients are actually asking you for, rather than simply taking anything they say and then moving in a straight line forever. And if both desires cannot be satisfied together, the best way to deal with that situation is to explain that that is the case (and why it's the case, if the client/manager would care) and then outline what outcomes are actually possible.