How do interest rates effect everyday people exactly, other than price inflation on goods and services (which is included separately in CPI)?
The only way seems to be interest rates on personal loans and mortgages. So if anything, we should only include interest rates in proportion to how many people are taking out major loans during the sampled period (and maybe some additional amount based on the effect on adjustable-rate mortgages, etc).
Blindly stacking interest rates on top of CPI doesn't really make sense as a measure of personal inflation, and "it feels like stuff got more expensive" (as a lot of other comments here argue) isn't so much an argument for this strategy so much as an argument that the CPI 'basket of goods' needs to be rebalanced in other ways.