I've heard that nonsense since I was teenager, making $5.15 an hour, the same pay that my own father made as well, except that his rent was $200 a month and he was able to purchase an acre of land for a few thousand dollars. When I was making the same pay at $5.15, my rent was $650 per month, and the same land he purchased for 3k or less would have cost me $30,000 at that time. There was no minimum wage increase to cause that. Even years later, the minimum wage was raised to $7.25 where it has remained the same for 17 years, yet the cost of living has increased by 55%.
Sometimes I have to ask myself if people actually live in the real world.
Another example, we live in two red states, we've looked into moving to blue states with higher pay and better healthcare options. I have extensively compared the cost of everyday living between the red states with low minimum wage to the blue states with higher minimum wage. The grocery prices are nearly identical, gas prices in the blue states are between $0.30 to $1 more per gallon (depending on the area), the rent in the blue states is easier to find and lower overall, the land acreage is cheaper in many areas as well. The only significant cost that is higher in those states, comes from purchasing a home and having a mortgage. Which could be countered by the higher pay those states have to offer or by simply purchasing acreage instead. On top of that, the blue states have better access to healthcare and more healthcare options.