Why didn't SkyNet just send the first Terminator back in time with a download of its operating system for safety?

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Because plot. In the Epic Rap Battle of History, with Robocop facing Terminator, one of Robocop’s lines was, “they should have made a time-traveling morning after pill.”

There are much, much better ways to kill someone if you have access to time travel than a murderous robot. But they wanted a movie with robots and the future, so the Terminator is what we got.