I don’t think it’s a correct observation, but more than that I don’t think it matters.

The claim itself is very vague (and I’m sure Grant Sanderson knows this). It could be fun to find connections between phenomena involving [math]\pi[/math] and circles, it could even be valuable, but it’s not universally enlightening nor even universally feasible.

The video linked in the question, showing a geometric proof of the Basel problem, is very nice and sweet. I don’t know that I would classify it as deep or illuminating. There are many proofs of the identity [math]\sum \frac{1}{n^2}=\frac{\pi^2}{6}[/math], some deeper and easier to generalize than this one.

Do they all “involve circles”? Meh. Why does it matter? They much more naturally involve the exponential function. Some would argue that the exponential function harbors the trigonometric functions and the trigonometric functions connect back to circles. Ok. I don’t see that this is a meaningful debate.

Ultimately, all manifestations of [math]\pi[/math] are connected in one way or another, so the question isn’t whether or not there is a path; the question is what is the most natural, logical way to organize things and find a useful perspective on what rests on what. That, too, isn’t a formal question; math isn’t a DAG.

I myself find it tremendously more natural to build things from the exponential function and into circles vs the other way around. This wasn’t the historical route, but historical routes are rarely efficient or logical.

If you consider, for example, proofs of properties of [math]\pi[/math], like its irrationality, or transcendence – I don’t know, I look at these proofs (there are many) and I see a hell of a lot exponential functions and not a single circle. Is there a “path” leading back to circles in a proof of the Gelfond-Schneider theorem? I don’t think so, and if there is, I think it’s at high risk of being convoluted and obscure. But maybe.

Another point which I think is missed by many is that they think the circle definition of [math]\pi[/math] is “elementary” while the exponential one is “complicated”. That’s not true. Putting the geometry of circles, lengths and areas on solid footing takes a significant amount of work, and once again it’s much easier to do by first constructing the reals, the complex numbers, the exponential function and the Argand plane.

Can this be flipped around? Sure. Is it better? Not in any sense that I can embrace.