I see. The thing you consider a jump in logic is assuming he went there to hurt someone. I'll elaborate. There are plenty of reasons to assume this (if you've been reading the facts in the trial), but the standard law enforcement uses for this "reasonable suspicion" is that Rittenhouse had an illegally purchased AR-15, obtained through a straw purchaser. Because the weapon was illegal, all uses of the weapon are illegal.

Leaving aside his plan to go to a car dealership (the owners of which did not feel the need to defend) in another state to defend it despite having no connection to it, leaving aside his lying to the police and the media, claiming he was a certified EMT, choosing to go to a protest with a huge, illegal weapon loaded with full metal jacket ammo (so bullets will continue to travel after hitting an animal or human target), the facts do not support his claims of self-defense. The only tenuous claim he makes that has any basis in fact is when a man briefly pointed a gun at him, after Rittenhouse had already opened fire. The man he claims pointed a gun at him did not, in fact, do this until after Rittenhouse shot him in the bicep, at which point he lost control of his arm and it flopped down, briefly pointing the gun at Rittenhouse. The kid showed up to cosplay Call of Duty. He's guilty of murder.