Part of the issue is that they're not interested in catching criminals, they are interested in rounding up firearms.
These proposals come from some erroneuous foundational beliefs:
1) That it's possible to pass laws which disarm criminals
2) That taking firearms away from criminals is going to cut down on crime, and taking firearms away from the rest of the population A) makes them and those around them safer, and B) restricts the supply of firearms to criminals.
Each one of these points is false, but it doesn't matter.
If the city was interested in catching criminals who use firearms in the commission of crimes they could. The state could increase the sentencing for UUW by a felon from 2 to 4 years to 15 to 30, with no plea bargaining, no reduced sentencing, no early release. Most felons are serving concurrent sentences for whatever crime they committed plus the weapons charge. It's actually not that hard to catch and convict them.
One of the criminals involved with murdering Officer Clifton Lewis in December of 2011 was arrested for UUW that came to light during a traffic stop. He had a weapon in his vehicle. When they do drug busts, they usua;lly have enough evidence to also charge the gang members with UUW charges. They could incarcerate the people who are actually perpatrating gun violence. They choose not to.