I agree with everything you said - except the above. The original purpose of the minimum wage was for unions to keep cheaper competition out of the workforce. In the early days, that meant blacks and immigrants. These days, the unions are in favor of illegal immigration (even though it undercuts their own members) because they believe the illegals will eventually become unionized and start paying dues.
The minimum wage causes unemployment (temporarily) because unskilled workers simply aren't worth that much until the economy catches up and other prices rise enough to absorb the increase (i.e. the cost of a Big Mac). But that's the whole point - by the time that happens, nobody's life has been improved. The net result is zero or, as you say, even worse. If the minimum wage was $100.00 per hour, a Big Mac would cost about $40.00.
BTW I turned 16 in 1968. My first job at a Burger Chef paid the minimum of $1.80 (I think).
I have a brother your age. You turned 16 in 1968, I turned 24 in 1968. I was a young single man, driving a new car which was my second new car, I lived with my parents but I was helping them, not the other way around. I had had an honorable discharge from the Navy for three years already and I was enjoying my money. I made 500 a month! Things were so different then that people had decided that I would never marry! They thought anyone who was 22 or older should have a wife and at least one child already! Can you even imagine that now? By the way my new car had cost me about 3400 dollars, it was full size with a V-8 engine, automatic transmission, air conditioning and a radio, “loaded to the limit” for those days. You could only finance a new car for 36 months then.