I don't think you can tie minimum wage increases to increases in poverty since a good number of the increases were during a recession (which is probably part of how they got political support) and when they weren't they didn't seem to have much of an effect on poverty (if it was already trending down it kept going down, if it was flat it basically remained flat), but that would make sense if the adjustments weren't really doing much more than compensating for inflation.
And if somebody working at mcdonalds started making close to an entry-level bachelors-requiring position a more valid interpretation is that the entry-level position is underpaying. Or maybe that the bachelor's doesn't mean much (which many don't anymore). But I don't think many people would switch from an office job to mcdonalds if they were paid similarly. The degree in that situation allows you to not have as shitty of a job, gives you more room to advance, and probably comes with better benefits and more consistent hours. Not that I think they necessarily should have similar wages.
I just think that argument is similar to when people say welfare is awesome and they should just quit their jobs. Everyone knows that's not true or else they would actually do it.
I get that minimum wage jobs should be temporary, but they're not just filled with high schoolers and college students who live with their parents or in dorms covered by student loans (or their rich parents). A full-time minimum wage-paying job should cover everything you need to live and right now a full-time minimum wage job is going to pay you $15k a year. In some places the cost-of-living is low enough that the current minimum wage can work for that, but it seems that in most it's not, and that's a problem that needs to be addressed on some level, if not by an increase in the Federal minimum wage (though, I still think it should be at least $8.50-$9 currently just from browsing the living wage MIT site and clicking random places where I know there aren't expensive cities and given that it has been that in the past, adjusted for inflation). I don't think minimum wage should only cover food and cheap housing with multiple roommmates. The high schoolers and college students will get part-time jobs that ultimately pay them less.
If you think that economic situations are too variable to have a Federal minimum wage that can pay a living wage in every area of the country then I agree, but I think in those cases where it's not enough there should be higher local minimum wages and there should be some mandate or encouragement for states/localities to do that. I wouldn't argue against the concept of a minimum wage itself.
[And I'm using "you" in the general sense, not to any particular person...]