They have tried this with juvenile non-human primates! For example, Horner & Whiten (2005) tried this with 2-6 year old chimpanzees. Clay & Tennie (2017) tried

They have tried this with juvenile non-human primates! For example, Horner & Whiten (2005) tried this with 2-6 year old chimpanzees. Clay & Tennie (2017) tried this with juvenile bonobos. Neither group overimitated. They do play, but overimitation is probably underpinned by the ability/proclivity to infer Gricean intentions, which non-human primates lack. There is a strong normative element to this, as well as the play element you mentioned — I expect, as adults, we’ve all engaged in some form of overimitation as an act of conformity.