Other threads and Howard_Casto's remarks resonate like a tide, so here's another one.
I'll pick the beginning of one scene to illustrate why Terminator is ridiculous: at the beginning of Reece's interrogation they start from behind him, his laser scan almost center screen. How can they explain that? They don't. They ignore it.
Here's a more involved but realistic scenario: John and company know the future, let alone going back in time, is un-thinkable to the folks of the 80s. They would make a cover story. The killer is some lunatic. Reece is some guy who's motivated to help. He does. The cop Terminator knocked out and took his car can corroborate the lunatic dude exists and give his description. Reece is not arrested, but is sitting there with all of them brainstorming.
But the 80s was about the lone hero and conspiracy stuff. The above wouldn't do. Of course, the premise of Terminator is ridiculous. After even a ten or twenty warhead exchange, the world is toast - for centuries. No need to bother after that. Not to mention that someone in the field recently said that AI would just cause our blood to sour or the like as bio-tech will soon be within us.