If you look at the CPI in the website you posted you will notice it is cumulative, but the PCE right below it is the percent change from a year earlier, so we are not comparing apples to apples, which leads me to question the author's motives as well as his expertise on the subject.  
  

 The same stability the author talks about in the long run CPI can be found in the PCE.
 

 The FED. uses the core (excludes food and energy) PCE because it is a current measure of what consumers are spending their money on, whereas the CPI is a relatively fixed basket of goods.

Here is a long-term perspective from the actual beginnings of the two series.