According to this article, a Foucault pendulum proved Earth was rotating. I'm not sure it really proved it.
If Earth weren't rotating and a Foucault pendulum started in a state with zero velocity, it would keep swinging back and forth along the same line. If at its highest point, it has a tiny velocity in a direction perpendicular to the direction to the lowest point on the pendulum, then maybe it would have such a tiny deviation from moving exactly back and forth that a human can't see that tiny deviation with their own eyes, but that deviation would result in a slow precession of the pendulum and a day is so long that the direction it's swinging in would rotate a significant amount.
If you have a system where a particle's acceleration is always equal to its displacement from a certain point multiplied by a negative constant, and it's not moving back and forth in a straight line, it will travel in an ellipse which does not precess at all.
I think the same is not true about a Foucault pendulum. Its acceleration doesn't vary linearly with its distance along the sphere of where it can go to the bottom and the sphere of where it can go doesn't have Euclidean geometry. My question is can we really conclude from looking at a Foucault pendulum and the laws of physics that Earth is rotating?
Maybe if its initial velocity is controlled to be very close to zero, we can tell from its precession that Earth is rotating. Also maybe if its arc is very small, the rate of precession for any deviation from going back and forth that's undetectably small to the human eye will be so much slower than the rotation of Earth that we can tell from watching it that Earth is rotating. That still might not prove Earth is rotating, because if the pendulum has a tiny charge in the presence of a weak magnetic field, the magnetic field could also cause a slow precession.
Best Answer
The thing that Foucault did was not just to predict that a pendulum would undergo precession, but also to predict which way the precession would go, and how rapidly, depending on the latitude of the observer. (Although Cleonis points out in a comment and in another answer that ascribing all of this analysis to Foucault is a historical oversimplification.)
If you imagine a Foucault pendulum set up at one of the poles, with the Earth rotating underneath it, you should be able to convince yourself that an observer standing on the Earth would see the pendulum complete one precession cycle every day. Likewise a pendulum swinging in the plane of the equator would have no tendency to precess, and one at the other pole would (relative to the ground) precess the other way. The precession period works out to be $\rm1\,day/\sin(latitude)$, which at Paris is about thirty-two hours.
Many of the mechanical details of a Foucault pendulum are set up to reduce the contribution of the parasitic effects that you're thinking of. Foucault used a very large mass, to reduce the acceleration imparted by stray air currents, on a very long tether, so that individual swings of the pendulum are very slow and any intrinsic curvature or ellipticity is observable. He was careful that the material of the tether shouldn't "unwind" the way some multi-fiber ropes do, which would exert some extra torsion on the motion. And the pendulum was released by tying it to a horizontal fiber and then burning that fiber with a candle, rather than by cutting the fiber with a knife or having a person just give the pendulum a shove, to minimize exactly the sort of parasitic horizontal forces you're asking about.
Even then you might be able to explain away a single demonstration as a lucky fluke. The real strength of the Foucault pendulum comes when you repeat the experiment so that all these tiny parasitic forces ought to be randomly different, but the precession period turns out to be exactly the same, and then you repeat it in a city at a different latitude and the precession period is different by the right amount.