I would strongly disagree with "coherence should not make no difference to the argument". This is exactly what we mean when we say "a coherence length of X um" - you take all the spectral components and correlate them with their future selves and then look at something akin to average correlation.

Therefore the "coherence answer" is exactly and precisely the correct one. What you are doing is simply explaining it with different words. "The lack of coherence" of the wave as a whole is THE argument.

What your explanation is doing is applying a narrow wavelength filter and evaluating the conditions for each \Delta\lambda, which is absolutely a valid approach, but it is the same thing. Overall I like your explanation very much, but please be careful about statements like "not really the answer, at least not directly" as they only point at a misunderstanding of the coherence explanation.