That is not how I understand it.

The “short” version of my understanding:

“Centered sound” is all about sound quality – basically the amount of overtones you produce, the “richness” of your tone.

The pitch is only about the base frequency of the (hopefully resonant sounding) note you are playing. If you are sounding too sharp or too flat, you have basically two options to correct this:

1- adjust the tuning slide of your instrument.
2- lip the note up/down as needed – at the risk of loosing sound quality if you have to overdo it.