On the other hand, the government can have a great influence on the basket of goods and services used for the calculation by determining which products and services are inside the basket.
I'd argue that the CPI metric is more fundamentally broken, and doesn't even work with a 100% honest government.
CPI is based on a basket of consumer goods, but that basket changes if prices go up. People switch beef for beans, cars for buses, heating for blankets, etc. It's a self-referential loop.