See all the weights here: https://www.bls.gov/cpi/tables/relative-importance/2017.pdf
Housing prices go into much bigger bubbles than rent. If interest rates go up or down, it doesn't very much affect the average person's ability to pay rent. But it has a huge impact on how much the average person can borrow.
My understanding for the reason that housing prices aren't included in CPI is because nobody buys houses in cash. So including mortgage payments in theory makes sense -- but it' tricky -- because everyone's mortgage originated at different times and under different circumstances.
But they should include the sale value of houses in CPI, otherwise its pointless. Just because the vast majority of people buy houses with a mortgage doesn't mean the sale price isn't a real price
Edit: Shiller was wrong (I asked him about this point after a lecture about his work on historical housing prices).