Similar to @elbanditoroso‘s answer. As a random American, I think the three most important types of inflation to communicate to the public would be a) something to capture wage inflation for the poor, b) something to capture cost inflation for the poor, c) something to capture the “amount of bailout” that is occurring (i.e. impact on financial stability/distortion).
From that perspective, I think looking at the headline CPI number overestimates a) and underestimates b) given current circumstances (those relationships would be different in other circumstances). I don’t think we really have a good way to measure c), at least not in an “inflation rate” type of way.
The biggest issue I have is with b): I think, due to technical issues, rental costs have been underweighted in the CPI for a long time.
And then when interpreting a), I think a lot of economic analysts still look at the CPI as a good proxy for the wage level, and still look at the wage level as a good proxy for the wage level of the poor, and that’s a mistake.
But that’s just one view based on my own values and what I think is important based on common sense. Like @elbanditoroso said, you would want different measures to analyze different issues, and those different measures allow politicians to mix and match and obscure them.