Interest expense and interest rates, concretely paid by people like any other expense is not any more nebulous than tracking other expenditures.
We already know the interest rate and various categories of interest expense paid by population. It is just not included. Your rationale sounds very much like "we shouldn't include it because the inflation rate would be higher".
There is nothing mechanically weird about not ignoring a real expense the economy and people bear the cost of.
This isn't touchy feely, this is concrete costs people pay. Not tracking is purely political.
What is the CPI actually used for?
To set various levers at the central bank level to manage inflation and keep the economy humming along.
One of those levers is... the Fed funds rate... which in turn influences most other interest rates.
So you'd be incorporating interest into the measurement used to set interest.
Technically, they pay closer attention to PCE.
Anyway, recursive sums like that are nothing unusual; as long as the sum converges it's not a problem. Any negative feedback amplifier does the same thing.
For purposes of setting Fed rates, it makes sense to exclude it.
For purposes of measuring perceived inflation, it makes sense to include it.