The answer to that is proper policing force, not custom laws that ban individuals from begging. It's not a beggars fault if a non-harassing "got any change" makes you feel uncomfortable.
Why do people get to impose themselves on me, from advertisers to panhandlers to petition gatherers, repeatedly, day in and day out? I'm not talking about ambient noise or other people going about their business, chatting with each other and interacting how they want to interact. Why is it OK for me to be interrupted multiple times a day--especially when I'm in places where I have no other choice than to be--before I get to say I'm "uncomfortable?"
I realize that the law can and probably will be imposed unfairly and I disagree with the overly broad impact that it will have, but I can slightly understand the motivation behind the original idea.
The question isn't about someone's right to feel comfortable, it's laws targeting an individual instead of a class. A place decides no petitioning? Sure, fine (putting aside any civil rights issues). A place decides petitioning is OK, unless they don't like a specific petitioner? How does that make sense? If the petitioner is acting correctly, why should they get a ban? And if they aren't, why isn't their unacceptable behaviour categorically banned?
As long as they are not directly accosting you, how is this their problem? How do you expect an individual person to know that you have been already approached 8 times?
At least where I come from, the logic goes like this: You don't want to deal with the public? Don't be in public. YMMV.
The solution to your discomfort is for homelessness to be tackled meaningfully, not to have the police get extra authority to remove those people from your sight.
You're allowed to feel uncomfortable whenever you like. But that feeling is no basis to prosecute people.