If Minimum wage kept up to inflation then there would be no inflation to speak of for most. Also the picture that you paint of the good old times was not true for many, most especially white collar workers. They did not have a union to protect them. From the late 40’s to the mid 50’s many, many struggled to provide for their family’s. And many children’s piggy banks were appropriated so as to have a small meal on the dinner table. I know, I was a witness there.
Edit, This was the initial period where women first contemplated leaving their home to get a job other then housewife. A decade later it became common for women to find employment once again. Two provider incomes then became the standard for the middle class.