Clue #1: a minimum wage job isn't something you should live off of. It is expressly for teenagers and for folks who use it as a stepping stone or fallback until something better comes along.
Who says? This is misinformation/propaganda being spread. If you look at the actual bill that instituted the minimum wage in the US (the Fair Labor Standards Act of 1938), the law literally says the reasoning for setting the minimum wage is "Congress finds that ... labor conditions detrimental to the maintenance of the minimum standing of living necessary for health, efficiency, and general well-being of workers causes ..." and then goes on to list negative effects of not being paid enough to live. So yes, the law quite literally states that the minimum wage is something you're meant to live off of. (Feel free to read the law yourself on the Dept of Labor website.
This idea of "teenagers can do it" is only a ploy to make people complacent with low wages. Remember a teenager at 17/18 can easily be out living on their own and not have the support of family (for many reasons: family doesn't have ability to help, family has cancer and teenager needs to support them, family is crazy/insane/drug addicts, family is dead, etc.), and so even teenagers should make enough money to support themselves.
Clue #2: these jobs usually require little-to-no skill, and consequently do not bear the value of $15/hr at current inflation/valuation.
When the minimum wage was instituted in 1938, the many US jobs were in agriculture or simple manufacturing. I don't consider those jobs to be "high skill", but that doesn't mean they're not super important (without food, we die -- about as important as you can get! and manufacturing gave us the modern world, despite many of those jobs being just to screw the same bolt on over and over). So for one thing, skill does not equate with importance, and I think important jobs especially should be well paid.
Furthermore, have you seen secretary and human resources job these days? Also requires pretty low skill (mostly just typing and sending emails and filling out forms -- anyone who can read and write can do it, really), but look at how much these people make (in my area, you can get jobs in HR making upwards of $50k with only minimal experience, much above minimum wage). If we were going by your metric, these paper-pusher jobs should be making low pay and important jobs like farmers and restaurants that provide me food should be making more.
All of this is an aside from the real goal of minimum wage, which is that if you do ANY type of work for anyone, you're important to someone and should be able to support yourself doing that work. If you're not needed, why did the company hire you? I'm tired of this idea that companies are entitled to cheap labor; if your company requires effectively slave labor to exist, then how about we state the truth that your company is failing, not doing well, and maybe should go bankrupt due to mismanagement rather than keeping it chugging on the backs of the poor?
Clue #3: when you price human labor too high, automation becomes more attractive. There are already machines that can effectively replace fast-food cashiers, and are cheaper to operate and maintain than $15/hr people. There are also machines coming online that can operate the back-end of a fast food joint as well, which will also just come under the wire as being cheaper (but would come out ahead by being reliable, on-time, etc.)
That is going to happen no matter what because of corporate greed to always maximize profit. Even if we paid people $1/hr, at some point people would need to eat and sleep while a machine could work all night long straight, cranking out more widgets. We can't compete with technology.
What we instead need to do is have real discussion on what the future economy looks like when jobs are phased out by robots. Probably future jobs would be more creative engineering or artistic jobs that robots can't do and it will work itself out and the economy will keep moving on, but we will have a transition period before we get there and it will be different than what we have now. In this transition, we need to do the humane thing and help people transition. That means making sure people's needs are met as they go back to get training for jobs, whether they be more academic (engineering/science) or more trade level (arts and crafts, music, cooking, stuff that makes people happy and can make money). It's not anyone's fault that robots were invented and are taking over, so why do we hold it against them that they should have someone thru clairvoyance known of the impending robot takeover and planned accordingly?
Clue #4: sucks to say it, but no one owes you a living -anything, let alone a "living wage" (whatever that means). Safety nets and charity are for those unable to help themselves, and obviously for those among us in temporary desperate situations, but that's it. Meanwhile, if you are able-bodied and not mentally defective, then it is up to you to better yourself by any legal means possible.
This makes the incorrect assumption that people have control over job availability. I know plenty of people with experience in fields that are drying up (drafting is a good example) -- used to make good money and be steady pay, now with automation, there's few jobs to go around. What is someone with 20 years supposed to do? That field has no jobs left; I know a guy that is looking. He also can't get into a new field because they pigeonhole him: "Why are you applying for this? Your experience is in a different field". He's quite able-bodied and intelligent person but its not working out. He's taking evening classes to go more into computer work, but that doesn't happen overnight. What do you propose he do in the meantime?
We need more than just a simple safety net, but a system that makes sure you have the chance to get ahead (good wages help you pay for things like school for example), and when things like the economy shift, is there to help you transition to new work.