“Did they wave the cow over the head in the Bais Hamikdash?”
– The 2 Shelamin sheep with the 2 Loaves on Shavuous were waved while alive, but not over the head.
With Kapparos the actual slaughter (which is an integral part of the ritual) is the more possible “Kodashim BaChutz” (severe prohibition of slaughtering or burning sacrificial animals outside the Azara – Temple courtyard) issue. But again, the severe violation does not apply to species that can never be accepted in the Temple as offerings.
Some may argue that one of the reasons chickens were completely “frozen out” from the Temple service was to allow their future use in Kapparos.
Sometimes Minhagim have had to be discontinued due to pressure or concerns from the “outside world”. The practice of over turning beds as a mourning practice – mentioned in the Gemara is an example.