They simply can't. This is very different from closing down some "piracy" sites. In this case they need to mess with data on your computer. Do you want FBI to have a legal way to read, modify or destroy data on your computer? Even if you want (which I don't believe), there is no way that would work on international scale.

I remember a case from some years back when some group (some AV group or something like that) took control over a major botnet by taking control over domains botnet would use in future for receiving updates/commands. The algorithm was pretty sophisticated as it generated dozens of new domains each day and was trying to contact them on random. E.g. generate 100 domains, try to contact 15 of them, if none responds wait a day. Statistically the bot would get its commands in a 2-3 days (I don't really remember exact numbers so this example might not add up). Well, in order to stop the botnet you'd need to register all available domains out of those 100. So, they were doing just that for some time. They practically blocked the owners operating on the botnet. But they couldn't destroy it. Because that would require manipulating with data on random peoples' computers without their consent which you simply can't (legally). And I think it should stay that way. So anyway, they left it be and owners took control back.

But what can be done? You have to run your operating system updated. Having AV software (which is quicker in updates and act as a prevention) doesn't hurt either. And as this is not a real option for a some time, this is a good way. You destroy it in illegal fashion. But no official organization can do that.

Victims of piracy have better lobbyists than the victims of identity theft.

Dutch police infiltrated a botnet recently, and left a message for victims. No one was too happy with them and they weren't really allowed to...

because the actions required to do so are illegal. This isn't about replacing sites or controlling the C&C machines. That's relatively easy.

To 'take down' the bots in the fashion of the OP means you upload the AVG removal tool so that the bots grab it at some point in their update cycle and run it.

What you've done here is arrange to run code on the infected machines that the owner of the machine is unaware of. Code that is intended to alter software on that machine, and that is illegal in exactly the same way as infecting the machine with the bot virus in the first place.

As the machine's are in the main unidentified and spread all over the world, no one legal entity FBI, SOCA, Interpol etc covers them all within their jurisdiction, or can target only the machines with in it, so applicable warrants and permissions would be practically impossible to obtain.

Because it is against the law. Taking down a site used by a criminal (or someone they believe is criminal) is within their purview. Making unauthorized changes to the computers of an innocent 3rd party (even minor changes, like removing a botnet infection) is against the law.