> a "gain of function" virus would contain random differences, and would look indistinguishable from something "created by nature."
No, a chimeric virus created in a gain-of-function experiment would look extremely similar to known viruses, because these chimeras are created by combining elements of known viruses. It would not be 4% different from the closest known natural virus. Accumulating thousands of mutations throughout the entire genome takes decades of evolution. In the wild, that means thousands of generations of hosts.
A virus created in a gain-of-function experiment would also use a well-known backbone. It would not be based on some virus that nobody had ever heard of.