"I don't think I hear overtones."

You undoubtedly do hear overtones; it's just that no one ever drew your attention to them so you thought you could play without noticing them.

I have theories about what happens to piano playing as a result:
Many pianists pay no attention to bass parts or to sustained notes.
Some pianists hum aloud to mask those uncontrollable vibrations.
Some experience utter terror when playing a new instrument on which the overtone blend is completely different.

Imagining the piano without overtones is like pretending that lightning has nothing to do with thunder. While it is true that lightning and thunder can be frightening, they are also fascinating. I haven't yet had a child student who was not mesmerized by overtones.