I acknowledge there's a full spectrum between they points (e.g. painting, where anyone can do it, but professional painters have many skills developed over years to make them both more efficient and with a better outcome).
Perhaps the problem with the label is that it's being attributed incorrectly? I don't think it's saying the person is unskilled in general, just that the person is unskilled in this job. There are plenty of instances where people that are skilled in one area but unable to find work in that area get a job in another area they are unskilled in to make ends meet, and I think those jobs are generally for unskilled labor. In those cases, they likely are unskilled in the job being done.
"We are now hiring for X. [Insert some pay, benefit information here]. Apply online. No interview required. Start tomorrow/this week."
At that point, you really are looking at 'unskilled' or at least 'not specifically skilled'.
It's difficult to push for higher wages or better working conditions when your productivity after 5 years is only 5% more than someone that started a month ago, but that's reality for unskilled work.
I've never really liked "skilled" and "unskilled" label classifiers because it implies a relationship between compensation and skill. I'm aware you didn't say this but I see it far too often.
I've seen a lot of fairly low compensated work that I know for a fact I can't do. We need to stop dehumanizing people with 'skilled' and 'unskilled' labor. A lot of it just depends on what people choose to do and what demands happen to arise.