I watched part of the Rittenhouse trial. Here's my understanding of the 2nd amendment and "bringing a gun to a protest":

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The question "why did you bring your gun to place X?" simply isn't a valid one. The 2nd amendment says you have the right to carry it. End of story. Why you chose to bring a gun anywhere isn't of any more concern to the law than why you chose to wear boxers instead of briefs.

I'm no legal expert, but that is what I got from watching that trial. At one point, the prosecutor asked Kyle Rittenhouse why he brought his gun and the judge told him to knock it off.

At another point the prosecutor asked why Rittenhouse didn't talk about the shootings earlier, and was rebuked by the judge for being in violation of the 5th amendment. The question about the gun was treated in a similar way.

I don't necessarily agree with this, but that seems to be how the law works in the USA. The mere presence of a gun is not grounds for, well anything, actually.