I don't think you realize that "synthetic" oils and greases are synthesized from petroleum. They share the same properties as "petroleum based" lubricants otherwise they couldn't be used in the same applications.

You also can't make the broad statement that "petroleum based greases can deteriorate rubber". Some might but others don't. Petroleum based products cover too wide a range of materials to make that claim. Even saying "rubber" covers too broad a range of materials to make that claim. "Rubber" includes a whole host of elastomeric materials with vastly different properties. Natural latex doesn't stand up to much of anything well but, thankfully, not much natural latex is used in bicycle parts. Buna rubber, which is pretty common in bike parts, stands up to petroleum products quite well.