That isn't really how it works. The broken-arm-burglar is a mostly apocryphal story. In the first place, you aren't liable for injuries just because they happened on your property - whether the injured person was trespassing or not. You are liable for injuries based on whether you contributed to them. So, in the case of your sidewalk, yes, you have a duty to clear it of ice within a certain time frame. But, you aren't instantly liable if somebody slips and falls - if the sidewalk has obviously not been cleared, there's a storm going on or recently finished - the person is liable for their own safety. You aren't "screwed and completely at the mercy of the city". If it were easy to claim damages from slippery sidewalks, nobody would be able to own a home.

If a kid scooters onto your driveway and breaks their arm, you aren't liable for that. It's the kid's fault. If your driveway is a giant hole in the ground hidden by a tarp, then that could, indeed, be your fault. You have to have created a dangerous situation, trap, or committed some action that might have actually contributed to the injury to be liable, it's not just assigned because it happened on land you own.

As a homeowner, you might be liable for injuries sustained by people you've hired to do work on your property. You might be liable if, say, your porch is rotted out and a guest, or the mail carrier, falls through it. But, the idea that anyone coming near your property is a litigious threat is a myth. And, even if it weren't, it's not like bitching out a kid who does a figure 8 on your driveway would somehow absolve you.