It's ridiculous to design a system for the purpose of spying on its users, with users being aware of that. The only people that will be discussing illegal activities using that medium then will be those so oblivious and dense that they likely would have been caught any number of other ways without the eavesdropping.

Savvy criminals will migrate to darknets and other methods of communication that are more robust against eavesdropping.

Personally i think authorities should just give up on the idea of eavesdropping. With proper encryption modern communications can perhaps be intercepted, but never deciphered.

Honeypotting seem like a much better strategy for catching internet criminals. Why would you ever shut down a child porn site or copyright infringement hub when you can instead take over and catch gather loads of evidence for prosecution?