Come on, you realistically expect the police to handle every case like this?
Police departments that currently exist? Not in every case we'd be talking about, no. We evidently need something new, but that something new is more like a police department than mob justice.
This is no different from having a reasonable right to self defense to protect your life.
The claim that mob internet justice is "no different" than individual right to self-defense is so utterly ridiculous that it borders on not worth responding to. Here is a rather meaningful difference: when you're going to shoot someone, you can see them and know what you're aiming at. I guess you didn't think of that.
If you are being harassed online you should be able to do something about it
I completely concur. That's the point of what I'm saying. You totally should be able to do something about it, and that something should not require you to become a private investigator, politician, lawyer, judge, and security guard. Nor should it only be available to those with enough resources: time, money, knowledge, physical or intellectual capabilities, etcetera.
The earlier you take action, the more you cut off the really bad stuff.
This behavior pattern - acting before thinking it through - leads to what's called "flailing". Experts will tell you pretty universally that this is one of the worst things to do if you're being stalked and harassed on the internet.
What if what is broken is having inherent trust in the system to do everything for you?
Then you've engaged in a strawman. Nothing about what I've just said demands "having inherent trust in the system to do everything for you".