> At one point does policy targeting a statistical measurement render that statistic useless?
When the statistic is a proxy for a quantity of interest, and when policy exploits ways of manipulating that proxy without commensurate change in the underlying quantity of interest. My driving policy of targeting my speedometer does a fine job of controlling my vehicle's speed even though the former is just a potentially-inaccurate measurement.
In the case of the CPI (actually the PCE for the US Fed [1]), the quantity of interest is the "[nominal] cost of living." To the extent the CPI doesn't in fact measure the cost of living, the easy solution is to change to a price index that does measure the cost of living. Since living tends to involve buying and consuming things, a price index, suitably defined, can be a fine measure of this.
To the extent you are concerned about the exact ways in which the CPI is computed, independent measurements [2] generally reproduce the broad trends in US data, which in turn suggests there is no significant manipulation by the BLS.
