I think you are quite confused about what overtones are and how they are produced.
For many mechanical oscillators (the air column in a flute, a vibrating string or free reed), there are modes of vibration satisfying the boundary conditions. A vibrating medium has inertial and elastic properties that combine in carrying the vibrational energy. This medium has significant boundaries which cannot support both movement or force. The boundary conditions usually support not just a simple fundamental vibration mode but also several higher modes of vibration: you can access them on some instruments as "harmonics" or "flageolet" by targetedly dampening of the fundamental.
For a string instrument like a piano, those higher modes can be almost but not quite proper harmonics of the fundamental frequency: they tend to be somewhat sharp, particularly for thicker strings and more compact instruments.
A tuning fork has torsional vibration modes around its tapped base. While it can in theory support higher modes than its fundamental mode, those are very much "inconvenienced" by the fork geometry, the location of the tap, and the thickness and curvature of the vibrating tines and thus tend to extinguish rather fast. They still make up some amount of the initial "ping" when striking the fork but fade out much faster than the fundamental and thus are not a significant part of the onsounding tone. The higher vibrational modes on a tuning fork also are not anywhere close to actual harmonics of the fundamental, so the fundamental cannot "feed" them in the manner it may happen with string instruments.