There’s likely a bit more to it than just the “ancient rivalry,” there’s probably some more specific reason that Kong gets involved in Godzilla’s reign of destruction.
Not that we necessarily need that much of a reason for Godzilla and Kong to throw down.
In the end, for the viewers, the spectacle of the battle is an end unto itself.
But once that happens, the ancient feelings probably kick in.
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GODZILLA VS KONG "Team Kong Vs Team Godzilla" Trailer (NEW 2021)