According to the text book "The Science of Sound"
Rossing et al, a plucked string results in a vibration that can be considered the sum of all the normal modes of vibration of that string. Because all the modes have different frequencies they will quickly get out of phase with one another and the shape of the string changes rapidly after plucking.
When the string is plucked with a finger or other soft object (including thin picks) then the point of last contact that allows the string to vibrate freely is spread out over a longer period of time and this prevents the higher harmonics from forming as the string has already changed shape over numerous cycles and much of the energy input to the string has been dissipated in lower frequency content and out of phase vibrations.
Using a pick, particularly thicker (more massive and more stiff) picks, allows the full input force to be transmitted to the string over the shortest period of time so that higher harmonic content has a chance propagate without damping from contact between string and pick.
Related to this is location of the plucking on the string. If the string is plucked in the center then the resulting vibration consists of the fundamental plus the odd numbered harmonics. Even numbered harmonics are canceled out by the pair of pulses that are traveling up and down the string in opposite directions. Moving the point of contact farther down the string results in fewer fundamentals being canceled out but also in less string amplitude (volume). This is why picking close to the bridge results in the greatest number of overtones but requires more work from the fingers to get the same volume. And it's why, when I do use a pick, I usually use a very large pick made of cymbal brass.
