Yes, regular singers are also overtone singers, whether they realize it or not! With overtone technique, the vocal tract movements become very microscopic, especially towards the higher end of the harmonic series. One learns to make these tiny adjustments by a combination of 'memorized' physical control and finely tuned harmonic hearing perception. By 'memorized', I mean that gradually the adjustments become automatic and intuitive. Like riding a bicycle, sort of.
Generally, in reading polyphonic scores (moving fundamentals), there are 2 ways to proceed. The first is paying attention vertically to the numbers of the overtones in any given part. The second, is to focus on the horizontal melody created by the overtones. Both approaches are very important to master, though with practice, a skilled overtone singer usually pays more attention to the melody, and doesn't need to think in such a complex manner while singing. The 2 voices move along intuitively.
Of course, you can say that overtone singing places more of the emphasis on melody since single overtones are isolated (particularly with lower fundamental pitches, accessing more of the series). As for managing 'fatter harmonics' in regular singing, I suppose that is one way of looking at it. Especially desirable when blending vowels with other voices, and a certain warmth is desirable.