The pitch of a tone is fact more difficult to recognize when it lacks overtones!
A tone can also be recognized if the fundamental is suppressed or missing. If you take a recording of a 440 Hz A played on a piano, and sharply roll of the lower frequencies below, say, 1000 Hz, you will still recognize the note the same way. It must be that the overtones almost certainly help the ear and brain zero in on the pitch of a note.
In fact, the brain can even "hear" a phantom fundamental anyway, when it's actually missing. Quote from the article: "It is now widely accepted that the brain processes the information present in the overtones to calculate the fundamental frequency."
Some kinds of signal processing software that recognizes pitch relies on overtones.
The overtones of a periodic signal have a spectrum that has a bunch of more or less equally spaced peaks. That spectrum can be regarded as a signal, in which those harmonic peaks look like a periodic pattern. The basic idea is that if you see multiple peaks that are 440 Hz apart, that indicates a 440 Hz A. These multiple peaks confirms the pitch better than a single peak, in the face of noise. They also deal with the suppressed fundamental. Even if the 440 Hz, or even 880 Hz components are filtered out or masked by noise, the tell-tale peaks of the other harmonics being 440 Hz apart still betray the note.
It's possible to take the spectrum of a spectrum to look for such patterns in the frequency space. This is called a "cepstrum". One of the applications of this is to determine the pitch of a voice.