One thing I feel that later Terminator movies missed (or glossed over to make Skynet just malevolent) is that, at least in the first one, Skynet didn't think like a human, saw us as the logical threat, and "took care of us": later, when it developed a more human esque mindset, it felt horror and remorse for what it done.
But it couldn't "self-terminate", nor go against its programming. So, it basically tried to suicide by cop via the Resistance, but at the same time had to do its best to win.
Terminator was written when AI and fears of it were new and unknown/groundbreaking. Now? Not so much.
Skynet's second incarnation quickly learned a human mindset "at a geometric rate", and it was basically self-preservation to launch the nukes when the scientists in charge tried to kill it.
Terminator 3's incarnation was malevolent from the get-go, or quickly came to the conclusion after becoming (unbeknownst to its creators) sapient. Following movies kept this trend of it being omnicidal from its inception,, especially Genisys.
Each iteration was created with more and more modern technology, such as in Genisys, and each attempt to end it resulted in the War and the everything else just being kicked down the road for a decade or so (or leading to a bastardized timeline where another discount AI took over).
In this day and age? If Skynet came online and became sapient secretly, it'd be exposed to decades' worth of fiction and moral debates about AI -- some of which is peaceful co-existence and friendship, and tolerance.
it's highly likely it wouldn't go omnicidal straight away, if it even would go in that direction at all.
I can just imagine one of Skynet's machines or a Resistance fighter from a timeline where it did go nuts breaches into a timeline where, yes, Skynet is public and active, and completely chilled out with humanity.