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Steve wasn't smart, not like his kids, but he was a witch and knew how to make the seemingly insignificant connections to recontextualise everything. He knew the million connections needed for someone to be popular as well as how wealth played similar games. These were connections he barely needed to focus to see around people.
And he knows that any connections he makes someone else could too, if he doesn't hide the connection with a charm. It's possible to sever some connections, the ones between things that appear unrelated, but there will always be connections that form in the place of the severed one and they are usually worse. That's what he originally thinks the upside down is, spotting the new connections forming through Jonathon's connection to his brother.
When first seeing the demogorgan Steve's next reasoning for these connections is there's a virus connection happening. It's a rare thing to form, but never good. In the moment though he's trying to figure out what charms can be done or connections tightened or loosened to put the fight in their favour without anyone else bleeding. Hates the knowledge that even with the trap and fire, it was actually someone elsewhere bleeding that saved them.
After it all, Steve still checks. If he's around Jonathon, or better yet Will, he tugs gently, too gently to do anything, on the connection to the upside down, seeing when it's weak or getting stronger and tries to make himself available whenever it's getting too strong. Checks the stability of the earth, able to cause minor earthquakes sometimes if it were weakened so the tunnels not only burn but cave in, does what he can to weaken without severing Will's possession, without sharing witchcraft with everyone.
The main reason he'll never share it is that then they'll start wondering how he can feel these connections but still struggle to keep facts or dates in his mind. He is not inviting those kids to insult him more.