If you model only the first-order effects (supply and demand of labour as a good, only) then sure, price goes up, it seems pretty clear that demand will go down

If you model only the first-order effects (supply and demand of labour as a good, only) then sure, price goes up, it seems pretty clear that demand will go down. If you model the economy in more detail (with stocks and flows of money, and where money in different places has different propensity to be spent and so on) then you can start to talk about second or third order effects which may result in an increase in demand for labour if minimum wages rise. I think ideologically the problem is that simpler models align with "classical / neoclassical" thinking, while more detailed models - even though they may be better (of course this is not a given) - tend to be less convincing firstly because they're harder to explain, and secondly because they tend to incorporate more parameters (so you run the risk of "fitting an elephant"). Personally I have my doubts that complex economies are actually amenable to simple explanations that fit human intuition at all - perhaps we'd be better off treating the whole field more like the weather, fewer elegant theories / word explanations and a lot more computational brute force and observation. I guess a big problem with that is that economics is inherently political, and politics demands narrative.