It's not "standard neo-liberal" - it's standard classical and most post-classical (including neoliberal) economic theory, and it is supported by about 50% of the studies out there. It is hotly debated, and so I agree that he should change it, but you're entering into exactly the sort of debate I warned him about with respect to this topic.

You're also correct in that it is hard to establish a causal link... it is also hard to find any empirical causal links in economics or any social science, because the data is not amenable to the scientific method - it is not repeatable, not controllable, and generally results in this exact sort of situation where specific sets of data are seen to prove this conclusion or that conclusion depending on their framing.

As to Stewart, his problem is one of generality... his examinations look at general employment numbers, not deeply regional/situational effects that might be quite different, and while this all hearkens to Card/Krueger, lots of work subsequent to them have gone back and forth on the question, finding more than "modest" dis-employment effects, all the way to validating Card/Krueger. And that's the problem. The actual science is a shit-show... because you cannot take an empiricist approach to something like an economics question. The data is just not amenable to inference without the risk of massive skew, massive interpretive contamination, researcher bias, since it is neither controllable nor repeatable.

And the effect is different if it is locally applied, vs. federally applied, etc.

Considering "the national economy as a whole" is all fine and dandy, but distinguishing this from "not just private enterprise" is sort of bizarre, since no one studies employment effects across an economy without including state actors in the evaluation, and no one that I've read, who is opposed to wage price-floors are unconcerned about the poor. It's just that they don't believe that minimum-wage controls are the best way to actually help the poor. So adding all the deleterious effects of poverty (which I 100% will stipulate to) doesn't add anything to the argument, it just seems like a vague way of making opponents of minimum wages sound like terrible people.

At any rate - even moderate disemployment is a big deal, since it is the least-privileged "marginal" worker who is most going to feel whatever disemployment effects might be found. So even a modest impact needs to be evaluated against other policy changes which might address the same concerns about poverty without disemployment, such as reducing the payroll-tax burden of employees, increasing income tax minimum deductions, or whatever they're called (basic amount under which you don't pay taxes), or other such measures which decrease or maintain relative cost of labor while still getting marginal workers more of their own money. And we could comfortably fund that by stopping bombing so many brown people in the middle east, or stopping giving money to agriculture and energy sector incumbents.

At any rate, even the bad effects you're concerned about (which I share concern about) aren't actually justifications of the policy, absent consideration of alternatives and of side-effects. That something must be done doesn't imply that any policy that we might imagine should be that something.

Anyway - his praxeology isn't complete... it's just not applicable to the sort of analysis you hope to achieve... which those who tend to appreciate praxeology argue is an unfit method for the study of social sciences such as economics.

David Gordon did a pretty reasonable overview of this methodological dispute in a video for Mises University.