If a true "minimum" wage is also the average wage, then it is also the de facto maximum wage.
If the average wage is the minimum wage, the highest wage is also the minimum wage - by definition. No wage can be lower than the minimum and thus if any one person had a higher wage then the minimum would be below the average. That means everyone would be earning a minimum wage, or everyone is equally poor.
It wouldn't surprise me if that indeed is utopia for some people.
There seems to be some confusion about the claim that if the minimum wage is the average wage that it is also the highest wage. Consider an economy with three workers, making 3, 4, and 5 units of currency respectively. The minimum wage (effectively) is 3 and the average is 4. If the minimum wage is set to be 4, then either the 3 will increase to 4 or stop working. If the wage increases, wages are 4, 4, and 5. The average is 4.33. If the job is lost, the wages are 4 and 5. The average is 4.5.
Now, if the minimum wage is set to 4, everything is fine and stable. But if the minimum wage was set to the average wage, the minimum wage increases to either 4.33 or 4.5. Same decision. The jobs not paying the increased wage can either increase or stop. So wages might be 4.33, 4.33, and 5 for an average of 4.56. Or wages could be 5 for a trivial average of 5. Or somewhere in between. After repeated steps of this, wages go to 5 for everyone still working.
This is just a mathematical tautology. If the minimum and average are equal, then the maximum has the same value.
When will an underground labor market develop? At what level will popular support diminish or disappear, if ever?
When a substantial portion of the laborers are priced out of the market by the minimum wage. You see some complaints now about high schoolers being priced out, college interns being priced out, retirees being priced out ... etc. It would not surprise me if illegal immigration is in fact driven in part by the labor shortage caused by lowly skilled natives being priced out by the minimum wage laws. If it is indeed a material factor, you can see the wave of anti-government, anti-illegal-immigration sentiment as confirmation of that underground labor market.
You mentioned also inflation. that's an interesting topic and rarely explored in the academia. Minimum wages have traditionally not been a meaningful driver of inflation, due in large part to their being a small component. If a larger and larger portion of the labor supply comes from minimum wage earners, and minimum wages are tied to inflation, you can see a multiplier effect there where run-away inflation is quite possible -> higher minimum wage -> higher inflation -> higher minimum wage .....
The potential breakage there is the employers decision, collectively, to automate, thus diminishing the demand for minimum wage earners.
As someone argued in those minimum wage debates, the persons who got hurt the most by minimum wage laws are precisely the group that the minimum wage laws intend to help.