They do this sometimes. You don't hear a lot about it, because it is covert, and nobody makes a stink about it in the headlines. But it is time-intensive and expensive, so you can't do it all the time to every target of interest. If you believe the servers already have all the evidence you need, and you can get the servers, it makes sense.
It's like the difference between hiring a private eye to shadow someone for a month, and simply requesting a subpoena. Both have their place.