There's a nice example of why people dislike proof by exhaustion on the Wikipedia page. The problem statement is "prove that all years in which the Modern Olympics are held are divisible by 4". One can prove this by induction by showing that the first year, 1896, is divisible by 4, and applying the fact that all Olympic games are 4 years after the previous one. Alternatively one can prove this by exhaustion by listing out all the years in which Olympic games were held, and checking that they are all divisible by 4. Naturally the first method is preferable, not only because it's simpler, but also because it proves the statement indefinitely into the future.
Only problem with this is that there's going to be an Olympic Games in 2021 thanks to COVID, and 2021 is not divisible by 4.
What is a good replacement for this example? The obvious thing is to choose some well-known, reliably repeating event, but it's not easy to think of something that isn't annual. There is Halley's Comet, but the period of that is not a nice round number (75.32 years).
NB: I notice the Wikipedia article has changed to specifically exclude the 2021 Olympics, but the more exceptions are listed (the cancelled Olympics during WW1 and WW2 are already listed) the less elegant the example becomes, too.