That's because minimum wage isn't actively pegged to a cost of living measurement, it's manually adjusted from time to time (ostensibly to be inline with cost of living). That manual adjustment is slow and decoupled enough that there's no runaway feedback loop.

If you legally define the minimum wage as a formula instead of an absolute number, and that formula includes cost of living, you can create a runaway situation.

The fact that the minimum wage isn’t pegged means it IS adjusted: downward over time due to inflation. That has massive societal impact as it reduces wages for folks over time, risking homelessness and other effects of poverty. It also reduces productivity growth. THAT has real risk of runaway effects due to the compounding effects of poverty.

I don’t see why a simple adjustment for inflation or cost of living is bad and risky a priori but allowing it to decline over time is a priori good and not risky.

There is substantial bias in that assumption.