if rents rise, and other CPI items also rise, then it's not true that this is "all else being equal". If other items also rise, it means that there's more money being spent, which implies that people have more money to spend, and thus, income must be rising.

If rent rises are offset by costs of other items going down, then CPI won't rise - and this would happen if the rent is consuming more proportion of income, and thus, less is left available for the other CPI items.

I dont know which situation we are currently in today. I suspect that income is rising, and thus, inflation is high. Increasing rates would push up rent, but not push up income, and thus the proportion of rent vs income grows higher and thus lower inflation (in the future).