Depends on your interpretation of time travel and what you think Skynet knows. There is more than one way to look at it. The movies themselves can't even decide which form of time travel to stick to.
In one funny interpretation where time loops around on itself directly, no John means never any need to send a Terminator back to kill him/sarah. No need to send a Terminator back means Cyberdyne never got the t800 and Skynet never existed. Neither did John, there was never a reason to send Reese back to bang Sarah. In this interpretation, the solution was always to let Skynet kill John. They can't exist without each other. The paradox is somehow broken by some outside influence that causes John's death, and the loop disappears. Blip. Life goes on, Sarah is still a waitress looking for a date, and the most shocking death of August 1997 is Princess Diana.
In another interpretation where time doesn't necessarily rely on the feedback loop, changes only affect the new timeline going forward and remnants from previous timelines are now sort of orphans in time. Sarah is dead, John never happens. The 800 remains. I'd guess Skynet probably has some knowledge of its creation, knowing from Cyberdyne's files that they studied from an 800 in 1984. So maybe it has a secondary objective to just lay down in cyberdyne and shut off, right? Or maybe it just goes on a killing spree with the basic objective that any amount of dead humans is a good amount and gets captured, leading to Skynet without John regardless. It ensures its own creation and the result of the war is anyone's guess. Probably not good.
Another way you could run with it is sort of a hybrid result of these two. Sarah & John are out of the game, and the 800 just lies in a ditch. Jumps into freshly poured concrete maybe. Starts selling curtains. Whatever. Its programming is ignorant of Skynet's origin. No other goals, just chill till the bombs drop. But Skynet never comes, because nobody got their hands on this sweet bod. Whoops, no John and no Skynet either. Whatever strays were sent back by Skynet from the previous timeline are just orphaned in time without a purpose. What now? Well, Cyberdyne wasn't the only company working on AI. Plenty of other fish in that sea. Cyberdyne may have had a head start but give it another couple decades and now you've got Google SmartNet up your ass instead. Time for someone new to pick up the slack on behalf of humanity this time.
Nope, T2 explicitly confirms an unchangeable future by making the T-800 from the first film the reason Skynet exists. As Dyson says, "Things we would have never..." in reference to the T-800's CPU and arm. What the characters say doesn't matter, because the characters are wrong, and are telling themselves what they hope is correct. It isn't.
While the first movie definitely operated on what was assumed to be a closed unchanging loop, the second movie does try to leave it more up for debate. It implies that they're still fulfilling the same bootstrap paradox by relying on the 800 R&D, while simultaneously promoting the idea that the future is not set and that Cyberdyne did not experience these attacks on the path to 1997 in the previous timeline.
We all know about the filmed ending that would have confirmed that the future indeed wasn't set and that Sarah and John had changed it. NO FATE. The final ending was changed to be more ambiguous to allow the viewer to ponder the question without a definitive answer. It's more interesting that way.
Personally I had always considered the events of T2 part of the known bootstrap cycle like you anyway, however the upcoming Dark Fate clearly interprets T2 to be a timeline altering event rather than a loop confirmation. 1997 was apparently pretty chill. So it looks like we have a new answer.
T1 was definitely written to be a closed loop but sequels reinterpreted that. It was inevitable, there's only so far you can stretch the bootstrap for sequels until it just gets unmanageable to write around. T2 was already stretching it thin.
Can T1 & T2 in isolation be viewed as a closed loop together that ultimately leads to Skynet? It can, and that's how I have viewed it for many years. Same as you. But is that answer definitive fact? No, it was intentionally left open for interpretation. Certain events in T2 were definitely feeding into the loop, but those could have also been very close calls narrowly avoided thanks to more time travel intervention. One could argue that while sending the first 800 back fed into a loop, sending the 1000 back accidentally broke it.
What's interesting to me about this interpretation (that T2 starts a new future) is what it implies about prior John. A history we've never seen. It means that the Future War John we briefly saw never experienced the events of T2. He never knew Uncle Bob or attacked Cyberdyne. Interesting to think about how
that John's life went, then. Before the T2 timeline. His mom rotted away in that cage and he kept believing she was a nut. He kept on living in Southern California with his foster parents being a little dick.
Judgment Day hit when he was 12-13 years old. How could he have survived that? Sarah must have eventually broken out anyway (you know she could and would) and kidnapped a very uncooperative John, who still believed she was a loon all the way up until the bombs dropped. This time it wasn't until the bombs that he realized the truth about his mom and how hard it must have been to keep him safe against his will. Thanks, mom.