(the average person doesn't care nor are they affected by how much their employer spends in healthcare)
It may be true that the average person doesn't care how much their employer spends on healthcare, but it's not true that we aren't affected. From an employer's perspective, healthcare, wages, and all other benefits are part of the cost of having an employee. When healthcare goes up, it increases the total employee cost.
Employers can handle this in several ways. They could reduce the amount they give investors (as dividends, stock buybacks, etc.). But then the stock is worth less and they have to make up the money somewhere else. They could pass the expense on to customers. But then the loss in business can easily cost more than the revenue raised. They can cut wages or other benefits. Then the average person will start caring...and might get a different job.
(I found this article saying that 12M households spend >=50% of income on rent, so I'm assuming that an even greater number spend more than the recommended 30%, which means rent should be weighted as high as it is in CPI.)
According to the census, that's only about 10% of households. It also notes that 64.4% of households are owner-occupied. They don't pay rent. The CPI makes up a number called owner's equivalent rent for those households to get to the higher percentage.
The CPI is intended for things like wages. This makes it a good choice for a cost of living adjustment, but it doesn't quite represent the overall economy. And for investments, it's the broader economy that matters. Household consumption is less important.
What the Fed says.