Most of it, yes. If they report that CPI rose by 1 %, say, then it is certainly true that the index rose by 1 %. Whether that index represents what people are experiencing is another matter.
I cannot comment specifically on the US indices because I am not familiar with the detail of their construction but I know from reports from US financial commentators, who I respect, that the indices in the US are manipulated just as they are in the UK and the EU. For example Shadow Government Statistics*
continues to calculate CPI using the rules that were used in the US up to 1990 and rules that were used up to 1980 and compares them with the official figures. Their 1980/1990 based calculations better reflects the price inflation that real people are facing in their lives. When the new rules were adopted in both cases they produced virtually the same results as the old rules but they have steadily diverged ever since. The level of divergence is a measure of the amount of manipulation. In the case of the 1990 based inflation the index is more than double the current version and for the 1980 based calculation it is around 5 times more.
In the UK the government changed from using RPI to CPI on the grounds that it is a "better measure". In reality CPI is virtually always lower than RPI, except in the short periods of time when house prices are falling. They are almost always rising in the UK. CPI excludes housing costs while RPI includes them. For the average person housing costs are the biggest single item in their monthly spend and housing costs mostly increase more rapidly than all other expenditure**. That alone allows the government to report a lower price inflation rate than people are experiencing in their everyday lives. For example the cumulative increase in CPI from 1988 to 2019 is 130 % while the cumulative increase in RPI for the same period is 180 %. It should be noted that the difference in the UK is not as big as in the USA because RPI was also manipulated in the same way as CPI during that period. The US comparison is between manipulated figures and unmanipulated figures. They also changed the basis for pension increases from RPI to CPI*** so it also means that pensions do not keep pace with real price inflation.
* Some economists do disagree on the validity of the Shadow Stats figures. They don't criticise the calculation as such but believe that substitution and discounting for higher performance is more valid.
** Energy might have higher price inflation in times of crisis but it is in both indices.
*** To be fair the UK State Pension now has what is known as the triple lock in which it is increased by the higher of earnings growth, CPI or 2.5 %. Going back to before the Thatcher era, UK State Pensions were increased in line with earnings growth but Thatcher altered that to RPI. Over a long period of time accumulated earnings growth was bigger than RPI so pension incomes were slipping backwards relative to workers incomes. The amount each year was small but the accumulated effect became large. The triple lock was introduced in 2011 to try to reverse the impoverishment of pensioners but many private/company schemes changed their increases from RPI to CPI without the triple lock so pensioners on these schemes are worse off. Whether they could do this depended on the wording of the rules of the pension scheme. If the rules explicitly stated that increases would be based on RPI then they had to maintain that. If, however, the rules stated that the increase would be based on "official rates of increase" then they could use the change in State Pensions to justify a switch to CPI. I have three company pensions and they all switched to CPI.