How did we go from asking what is the proper wage to suggesting its "not a real job"?

All jobs are real but some are less valuable than others. There is nothing wrong with being a grocery store stocker your whole life but I am not sure how you set the wage for a 5+ year worker that never moved on/up from stocking shelves. I think its a fairly difficult problem to answer and throwing in classism and white collar vs blue collar is just deflection on accepting its hard to come to the right answer here.

A really long time ago, I had once visited a police station in Tamil Nadu (India). One of the lower level officials there had the rank "SSI". I was perplexed as I had never heard of this rank before. So I asked the Inspector about it. He said it stood for "Special Sub Inspector". He then smiled and added, "There was some court case involving some government official or union who complained about not being promoted even after serving for many years in the same post. The court ultimately ruled that was unfair, and said even if an official doesn't get promoted by virtue of merits, their experience also matters. Thus, any official who serves a certain number of years (I think it was 5-10 years or so), and doesn't get a promotion on merit during those years, should automatically be promoted by virtue of their experience. Ofcourse, some government bureaucrat in the Tamil Nadu civil service and police service weren't happy with this ruling as police officials who don't get promoted don't because they are either corrupt and / or incompetent. Thus, the government created a new rank (in between the ranks of Head Constable and Assistant Sub Inspector) to "comply" with the court ruling but cheekily called it "Special Sub Inspector" to convey their own disagreement with the ruling."

It was funny but it did make me think - Perhaps in the mid-levels, experience can compensate for the lack of special skills or motivations as an employee will know enough to do his immediate superior's job "well enough" (average or slightly below average) too.

I don't think answering that problem isn't at all especially if you feel human dignity is a value that should be honored above all other rights. And I don't think many people disagree that to work a full time job and be unable to live off it in the area it is located in is undignified.

Now markets have shown time and time again that precisely that human dignity is ignored within market logic (just as other externalities like air quality, a unpoisioned environment or the rights of future generations). Given the markets inability to honor human dignity I suggest regulating this part of it by mandating a minimum wage that is a living wage and keep that wage updated at least on a yearly basis. Not a radical thing to demand.

A real job is one that allows you to lead a dignified life. Being okay with essential jobs being paid below that wage is a problematic stance to take for various reasons and comes across as an egoistic stance to me (e.g. imagine someone saying: "While I enjoy cheap and good coffee, I don't want the batista who made it to afford to live off it.")

There are not many reasons to say that, but all of them are to some degree egoistic. You might be a star bucks manager that makes a good living off your workers earning less. You might be a customer that wants to pay as little as possible and is okay to ignore the exploitation. Or you might think this is market logic and go like "they simply have to find another job" (thus admitting it is not a real job anybody should expect to live from).

Now my stance — just like yours if I got that right — is that all jobs are real jobs.

That 5+ year grocery store stocker is the lynchpin that allows them to quickly hire teenagers who are only going to work there for a year at most before moving on. Without the institutional knowledge of the long-term workers, the store would have to spend much longer training new employees, or employ a magnitude more supervisors.

This kind of role exists but not in the store stocker or dishwasher side. Those are for better/worse roles that take a few hours to learn. Maybe some things you pickup on the way but by no means a role where it helps to have experienced folks in.