That's not obvious to me.

It wasn't intended to be obvious, merely true. But (1) I'm merely repeating something I heard from someone I thought credible, so it might turn out to be wrong, and (2) as I recall, "not many" is still rather more than one ever sees in practice in actual games.

So, is it right? I can't readily find anything on the web where the calculations are done, and I'm too lazy to do them properly myself. What I can find is a good look at a related problem, which gets us at least part of the way.

So, the (excellent) book "The SIAM 100-digit challenge" by Bornemann et al provides solutions to ten problems of the following form. "Here is a mathematical question whose answer is a number. Find the number to at least 10 significant figures". One of the problems, whose proximity to the billiards question should be clear enough, is as follows. Put a perfect circular mirror of radius 1/3 around every (integer,integer) point in the plane. Fire off a photon starting at (1/2,1/10) with velocity (1,0). Where is it after 10 units of time? The answer isn't particularly important, but it turns out that (1) in that time it bounces 17 times and (2) the growth of any initial error is about a factor of 10^11. That suggests about a factor of 4 per bounce.

So, the next question is what quantum effects do to the initial trajectory of a snooker/billiard ball. The following is surely not the best way to answer that question, and I suspect it yields a considerable underestimate, but let's see. Consider the first bounce. A change in either the position or the momentum of the ball will alter the angle it bounces at; for balls of mass m and radius r moving at speed v, a change in position by dx will change the angle by something of order dx/r, and a change in momentum by dp will change the angle by something of order dp/(mv). So the angle uncertainty has to be at least something of the order hbar/(mvr), which is probably somewhere around 2x10^-31 radians.

So if a "substantial effect" means, say, 0.1 radians, that means ... actually somewhere around 50 bounces, which is distinctly more than the number I thought I remembered. So maybe the bit of folklore I heard was wrong; or maybe I misunderstood or misremembered it; or maybe a better estimate of the effects of quantum phenomena on a billiard ball gives a much larger value.

the three-body problem

... doesn't have a closed-form solution in classical mechanics, but that's an entirely separate question from whether it amplifies small quantum effects enough to be useful for generating usable quantum randomness. (I think the answer to that depends on the details, but in any case gravity is weak enough that any 3-body system you make is going to be made up of massive objects moving on rather long timescales; it'll be much less practical as a quantum randomness amplifier than a system of bouncing billiard balls. Of course both are ridiculous choices anyway.)