One pure sine wave does not have an overtone or harmonic. The sine wave is a single unit of information in some version of signal processing theory. It is a mathematical function.
The overtones or harmonics that contribute to musical instrument tone or timbre come from solutions to the wave equation, which coincidentally is also, in many cases, a sine wave or some modified version of sine wave. When you play any instrument you need to "attack" it to produce sound and that attack will necessarily produce overtones, unless the attack is due to sympathetic resonance with another source but even in this case the initial turning on of the source WILL have harmonics in it due to the leading edge of the wave. This is a fact of life that will never be absent.
As for the "pure tone". No such thing exists in the sphere of human experience. The mechanisms in the human ear are non-linear and generate aural harmonics when excited by an otherwise pure tone. So the brain will not be fed a single frequency when provided a single frequency driving force. This can be tested using expensive function generators and listening to intervals produced by a series of them. It is not fair to say that the theory failed in practice since in practice you always excite harmonics in your ear.
You would have to provide more information on what you mean by "... but then what’s the reason we feel these effects that are caused by the overtone series?" relative to a pure tone. Under what conditions do you think you have "felt" the overtone effects when listening to a pure tone or set of tones?
Other causes of this could be related to other acoustic properties of the room, earphones, instruments etc. Stiff rods for example produce overtones that do NOT follow the harmonic series and are quite dissonant. A stiff rod with clamped boundary on one end can produce an overtone that is (if memory is not too faulty) about an octave and augmented 5th above the fundamental. When struck with enough force this harmonic is very loud and can be perceived as a distinct note. It can also be picked up using a mic and a AD data acquisition card. I've experienced this myself in a controlled setting. A less known factoid is that metal strings in a piano and on an electric guitar can exhibit this stiff behavior especially when very short and under high tension. Coupled with the resonances of the body it is not impossible to hear some of the dissonance when one plays even a single note on an instrument.
In short (1) your ear is never harmonic free, (2) there is more to acoustics than the harmonic series, (3) unless you are in a controlled environment it isn't really fair to say that these effects are "perceived" when you expect them not to be. It is an oversimplification of physical acoustics.
Don't confuse "pure tone" with "I only played one note".