Leave it to the Great State of Chicago to write a pointless law that makes it as difficult as possible to comply with other than to spend lots of extra money.
Sure I will permanently alter (read Damage) my firearm so that I can go hunting whitetail deer.
First off, if you have to fire more than one round to take a deer you probably should not have taken the shot to begin with.
Second, I have never been hunting with anyone that just starts firing shot after shot at a deer. I have only heard one story from another hunter of someone he hunted with doing so.
This law pretty much eliminates pistol hunters. Unless someone starts manufacturing pistol cylinders with only 3 chambers for revolvers.
It's not worth researching, but I'm pretty sure the previous law eliminated pistol hunting, except with single shot pistols, like Thompson Contenders.
Now it opens up the field to the infamous three-shot pistol.
The law basically doesn't allow detachable magazines (if you plug one magazine, nothing stops you from using a different one). The only bizarre jump from one to three shots I can think of would be plugging a fixed magazine pistol, like a C96 Broomhandle. Or that weirdo (you know he's out there) who plugs three chambers in a revolver. The government says,"See, our benevolence is easy to follow."
The Thompson Contender has been around for decades.

Bond Arms added the Cyclops to their derringer lines.

If a centerfire handgun or centerfire rifle is capable of holding more than 3 rounds, it shall be fitted with a one-piece plug that is irremovable.
“If a centerfire handgun or centerfire rifle is capable of holding more than 3 rounds, it shall be fitted with a one-piece plug that is irremovable without dismantling the handgun or rifle or it shall be otherwise altered to render it incapable of holding more than 3 rounds in the magazine and chamber combined without dismantling the handgun or rifle.”
This allows for a plug to be placed in a tubular magazine, or below the follower in a magazine fixed to the rifle. There are quite a few bolt action rifles where the magazine cannot be removed without some disassembly. Those would be pretty easily altered to hold only two rounds.
Still bad, but not quite as bad as requiring an “irremovable” plug.