One big risk here, or with any cyclically referenced benchmark, is runaway positive feedback. This can be exponential growth, oscillation, or exponentially growing oscillation.

If wages are referenced to CoL, but CoL is affected by wages, you have this risk.

A real example I have encountered: years ago, before an area I was living in experienced hyper demand, rent prices were fairly stable. I was able to get decent places just under $1k. My landlord was pretty chill, but said that they had to raise the rent eventually because they couldn't write off expenses (or maybe it was losses) if they were charging below the area median rent. So every year, landlords would have to increase the rent to the median, which would almost certainly raise the median (unless literally the bottom 50% of prices were exactly the same), and then they have to raise again, etc.