You forget one point. Jill doesn't know who exactly he is going to help and who he will take care of. It could well have been Josie, with whom MС is friends, it could have been any girl at all, to whom MС has no serious intentions, but for Jill it's still a red line.
I don't actually know who you're referring to with "you forgot one point," but how does it matter to Jill who it is? She knows MC is dating Josy as well as others. The basic action and relationship dynamics are the same in this case. I'll go ahead and quote myself a few times here:
"And if a guy went to girl's place to "help her" when she's just a little sick? Hello, no. It's completely unbelieveable to think that this wouldn't be taken as choosing another girl over you. Seriously, who actually does that in real life? "Oh, I caught a cold while in the rain. Come over and pamper me.""
"Also, hold up. How many times have you gone over to a platonic friend's place and pampered them when they had a cold? Not the hospital. Not a car accident. Not some deathly ailment. A damn cold. If the answer's zero, then there's the real answer."
"I'll ask again, how many times have you been asked by a purely platonic friend to come over and pamper them when they have a cold?"
Seriously, it just doesn't happen. Sage wants the MC bad, arguably as a closer companion than anyone in HOTs. That's why she asks him to help her instead of anyone else. If you pick Sage as your love interest she confesses that the real reason she was torn about dating MC is that she values his relationship so much that she didn't want to risk losing him. This is a completely believable reaction, though it follows the overwhelmingly common pattern that "friends with benefits" almost never works out. People inevitably catch feelings. There have been actual psychological studies on this. And Sage very obviously has feelings.
But anyway, I'll push back again, as have others, that this singular action, out of context with everything else, is a red line for Jill. It's just as much a matter of what doesn't happen if you don't go on the date as anything else. It's the timing of what Jill decides in that moment (when Tybalt is blackmailing her and she's basically decided that it's best for everyone involved if they break it off). You don't have very important conversations and bonding moments, and time passes after Jill has made up her mind. All of this context is critically important. You can't just ignore it and have an honest conversation.