Do you really want to turn this political? Plenty of work - including factory work - something that was decidely a "career choice" (in fact for most of those most prosperous years of the US it was the primary career) has been largely staffed by minimum wage work for most of the years. (Although post-unionizing many of them did get pretty decent - but they were considered a valid career choice previous to unionizing even) - in the first half of the 20th century people would raise families and own homes on a SINGLE minimum wage job.
And considering how sparse jobs are in some areas, pick and choosing what's a "valid career choice" is a laughable argument - unless we're going to start paying to help people relocate or reeducate (which I'd wager would be costly - and quite possibly cause a lack of service industry employees that our economy is largely driven by - which would in turn cause them to have to increase wages to fight for a scarcity of employees).
Fuck, if you even look at the costs to most of these fast food businesses, payroll is like 20-30% of their actual overhead at most. As a consumer assuming they wouldn't tighten up hours a bit in response that means your $5 burger meal goes to around $6.
So much angst over trying to save a buck or so a day. (And before the "wah-wah inflation" argument, you might want to look at the actual evidence on how minimum wage and inflation has functioned in reality - which really shouldn't be shocking since it just disperses existing income, it doesn't actually create any - same pool of money with more people using it just creates more economic movement, doesn't increase the size of the pond - the NJ/NY study was really telling, and by the end of next year should have an update as well I'd imagine)
And before you start weighing in with "people don't do a better job with more pay" argument - yes they do - I've seen it firsthand in my own hiring back in my HR days. People that would come to my employer (who paid well in excess of minimum wage for "untrained" tier employees - I was allowed to start people at $11/hr without a HS diploma/GED ten years ago) working two jobs because they felt they still needed it, got their first raise or two started making enough to drop their second job and suddenly are seeing all their metrics increase drastically overnight, I can even recall multiple cases of someone doubling their metrics once they weren't burning the candle at both ends anymore. Sick and tired employees by nature do shitty work. As do employees that are using their brainpower to juggle two or three sets of company policies and procedures.
Additionally - right now welfare and disability pay people nearly the same amount FOR DOING NOTHING that they can earn doing minimum wage work - doesn't it make a modicum of sense that the people that aren't leeching off the tax payer teat see a substantially more comfortable life than those that are being taken care of via Federal dollars?
TL;DR - You're wrong historically, you're wrong logically, and unless you just expect to throw people to the wolves like some third world country - other solutions likely involve money that will impact your bottom-line far worse. Period. (And no, I don't do anywhere near minimum wage work - this wouldn't impact me in the slightest, except perhaps negatively, since I work in hiring people, some of which are factory level work that might consider $15 fast food over factory assembly with a slight danger level for a little bit more - before you even consider that nonsense)