A very simple way of putting this difference is that when you shift to a subjective standard, you're just formalizing the heckler's veto. That's what you're doing. You're giving anyone who claims offence and who claims offence and wants to express that.... There are some people who won't want to do it, but anybody who wants to claim offence is going to be able to do that and effectively gain heckler's veto, because they, in theory at least, have a claim against the other person for hate speech. They can get damages and other sorts of remedies for that under the proposed amendments.
The concern is precisely this: Instead of having some kind of objective standard that weighs both the expressive rights of the one party and the rights of the other party, you're one-siding the conversation. You're saying it's the offence that matters and we don't have to take into account the expressive rights of the person entirely and also the values they embody, like democratic participation and everything else.