It doesn’t matter what Skynet does.
John Connor already exists in the time when the first Arnold Terminator is sent back. Sending AT back prompts Connor to send Kyle Reese back, ultimately resulting in John Connor being born. He causes his own birth (freaky!). John Connor only exists because Skynet exists, and because they sent AT back to kill his mother, and the remains of AT were the basis of the research that results in Skynet, Skynet only exists because John Connor exists.
Here’s a mindfuck for you: we know that in the future Skynet exists, and John Connor exists, and this loop had to have begun at some point, so there must be a timeline where Skynet is created without the remains of AT, and where John Connor is born without Kyle Reese, meaning that the John Connor who sends Kyle Reese back cannot be Kyle Reese’s son. So, who the hell is that in T2?!