Exponential growth is pretty rare in nature, right?

Linear growth or fuzzy up and down growth/swings seems way more commonplace

Epidemics generally follow a logistic curve, like most "exponential" processes. It looks exponential at first, but at some point, it decays. COVID is no exception. It is now in "fuzzy up and down" mode.

True, unbounded exponential processes don't exist in nature. There is a simple explanation about why not: assuming the speed of light is the speed limit of causality and we don't have weird things like warp gates, then the influence of something cannot go further than a sphere around it expanding at the speed of light. The volume of this sphere of influence increase following a cubic function. Now, if an exponential process happen withing that sphere, for example a population growth, it will run out of space at some point, because an exponential function will always beat a cubic (or any polynomial) at some point. The fastest growth possible is cubic, not exponential.

The scale of COVID was rare. I guess viral infections though aren't, and they follow a curve like that, just on a smaller scale.

and yet plenty of us can expertly catch balls thrown in the air

By more or less holding its angle constant by moving. The intercept trajectory is nowhere near optimal.

That's quadratic plus maybe some drag terms, humans are decent at projecting ballistic arcs.