My experience is that empirical economists care a whole lot about methods. They care about methods because they help us to understand problems we care about, of course. But what fires them up is neat modeling tricks that address tricky problems.
I can only speak from experience, but getting to know professionals in three different economics departments, a policy department, and professionally at the Urban Institute (both Urban economists and those I interacted with from elsewhere), they would get a lot more excited about their argument than about their answer.
This is not the case for, say, an advocacy organization in D.C., or activisty types I've come across at school and professionally. They care a lot more about the answer. And they'll cling onto the literature that gives them the answer that they want. But then again these guys don't write up analyses.
The minimum wage stuff is actually a good example of this. I don't really support the minimum wage as policy (although evidence for zero effects makes me OK with states trying things out - they have a better sense of appropriate levels - so I'm not a purist in that regard). Even at the state level I think it's an unnecessarily blunt instrument. And before I read the literature several years ago I personally thought that the minimum wage must have disemployment effects. At the time it fit my libertarianish ideological requirements, it fit what I thought I knew about economics, etc. I've really come around to the alternative view on the disemployment effects not because I was some huge energetic supporter of the minimum wage, and not because it was consistent with my ideology at the time, but despite those things - because the evidence was good. If there were disemployment effects in some cases (which I still expect there are), certainly there were enough positive employment effects elsewhere to get an averaged effect of zero.
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