We should make choices based on communication, first and foremost. Words that do not communicate are not useful.
A "quick test" implies almost nothing, other than it is faster than some other, unnamed test out there. Is a quick test good or is it bad? No way to tell. This term, quick test, it does not inform.
On the other hand, a sanity test, well, it is more evocative. You definitely want to pass a sanity test.
If you reach back to your psychology classes, they talked about a four-part test that is useful for determine if a behavior is sane or not.
1) Is it abnormal? Unusual? Out of the ordinary? for the situation.
2) Is it unjustifiable, unreasonable given the circumstances?
3) Is it counter-productive? Which is to say, does the behavior serve the individual, or does it in fact make things worse? Or simply do nothing?
4) (and this is where my memory is fuzzy) I think it involves personal distress. For the life of me I can't find it via Google and it annoys me.
Now, interestingly, a sanity test matches the first three! (The fourth, well, there is no "I" in the program to be distressed) In programming, a sanity test looks for something abnormal, not reasonable for the program, and represents a state that won't get us the desired output. In short, it's highly congruent to the other context for sanity.
This term, it communicates, and in a way "quick test" does not.
In general, one of the main critiques of political correctness which few really notice is that the new term is less specific and less useful, that it fails at communicating. Consider when "Oriental" fell off of the euphemism treadmill for "Asian." We all knew that the first term referred to a particular part of the world, from the Latin for "east." (Ex lux orient and so on) "Asian," however, could refer to people in Russia or people in India, but nobody in the US does (I note in the UK India does get the "Asian" pass). So here the new term is less specific and less useful, and is confusing to boot because we deliberately ignore parts of Asia when using the term "Asian."
If you want something to supplant "sanity test," you gotta work harder for it than "quick test."