I find fault with your original premise: the world shown did not look much like Earth of the 1960s. If anything, it looked like several decades before (the cars looked 1940s and the streets and buildings looked like the generic hollywood backlot they were).In the episode Miri we are shown a planet that looks just like Earth and in the same era that Trek was being filmed. Did it have a real life parallel? Were there any experiments going on in the 1960's that attempted to prolong life using viruses? Was the writer warning that man shouldn't be attempting this kind of thing?
That said, I think the warning was more broad: the law of unintended consequences. The things we do always, always have effects we did not plan for. That is why it is best in all things to proceed with caution.
That and the fun of a "what if" story. What if someone trying do do good had things go horribly wrong? What if someone was doing genetic manipulation of viruses without proper safeguards, and it got out?