Why would a publicly and a privately traded company react fundamentally different to the economy? What macro factors are you referring to, that fundamentally influence one but not the other?

For one, selling shares in a private company is much harder (if not impossible, barring buybacks), so that's why valuations are relatively stable (and ideally only go up, until an IPO). So by definition, private equity is less volatile than the public markets.

Secondly, private investments are a fairly well-studied hedge against inflation (due to return horizons being relatively low and return multipliers being relatively high) and also do well during, and immediately after, a recessionary period (where public markets generally do poorly).

(But maybe I misspoke, they are both, of course, affected by macro factors, but the effects are probably much more attenuated in private markets.)