A couple of points:
First, Chicago's problem with guns is not because of their gun laws. The gun problem in Chicago is fueled because Indiana and Wisconsin have such lax ones.
Boy that is making a big leap for causality.
Do Wisconsin and Indiana have gun violence issues comparable to Chicago? From the Wikipedia article using 2010 data, both of those places actually have LESS gun violence than Illionois, and way less than Chicago.
I suspect the gun violence in Chicago is impacted by things not related to gun laws more than it is to its neighbors' gun laws. Perhaps income inequality, social mobility, and education. Gangs are pretty big stereotype for Chicago, and I'm betting they get their funding from the drug trade. Perhaps that has something to do with it?
To state as a fact that Chicago has high gun violence because its less violent neighbors have lax laws just sounds like a big leap to me.
Second, your assertion that there's a large population that wants to "get rid of" guns isn't true. Most people want to have a dialogue on sensible laws. Not ban them. Those who want to ban them are a tiny fringe of the population, and leading people to believe otherwise is just perpetuating the polarization.
There absolutely is a group of people for whom that is the end goal. As I said, many of them RIGHT NOW want to make "common sense" gun laws (many of which are actually ridiculous). The problem is, there will continue to be gun violence no matter what laws you put in place.
I challenge you to ask your typical pro gun regulation friend what an acceptable number of gun deaths per year is, or mass shootings. If their answer is "0 of course" then they will want to continue passing laws until we reach 0, which is never. During every mass shooting media coverage a ton of people start shouting for a ban on "assault weapons." Once those get banned, and the next shooting is by a pistol, guess what they'll want to ban? Then when the crazy guy shoots up a kindergarten with a pump action shotgun, guess what'll be top of the ban queue?
There are already states that ban guns based on arbitrary "features" like a magazine release or the number of rounds in its magazine. Some of them ban individual guns by name due to how a committee thinks they look, or based on the muzzle device at the end of the barrel. We already have laws in the books banning certain barrel lengths combined with stocks. Take a look at the attached picture for some arbitrary rules determining whether an AR-15 requires an NFA stamp or is a felony. I'd LOVE to see the research that shows how only allowing angled foregrips instead of vertical ones reduces gun violence in any way, or how a 15" barrel being illegal and a 16" model that's identical being legal makes anyone safer. Until the pro "common sense gun laws" folks show they'll get rid of regulations that are ridiculous, I see no reason to continue compromising toward what they want. Why else other than "well at least it gets rid of some guns" would the rules outlined in the picture be in place?
Even in the era of OSHA, silencers/suppressors designed to protect shooter and spectator hearing require a $200 tax stamp, and approximately a YEAR of waiting for the extremely pain in the ass paperwork to go through.
Many gun owners I know would be fine with adding laws that might reduce gun violence, IF we get rid of the laws that only make things a pain in the ass for us and don't help anyone. For example, give us mandatory classes, IF we can buy a silencer at a neighborhood gun store like any other accessory. Create a federal waiting period of a few days, IF we can buy short barreled rifles the same as long barreled ones. Let us import guns from currently banned countries (who we still import all sorts of other stuff from), and we can do private party sale background checks. That's what I call an actual compromise and "common sense" instead of the current model where we always compromise toward more laws.
The current "compromises" are like if a racist said "I want all black people kicked out of the country, and you don't. Let's compromise and only make it where they can't drive cars." That's not a compromise at all.