Or they could take it down for enough time for other parts of the government and/or international diplomatic system to do their work.
Remember, a data leak is not just a technical issue. They can resolve it in any number of ways - get a small team incursion into another state's territory for extraction, etc. All the outage needs to do is to hold open that window for enough time for all the different parts of the entire threat response chain to do each part's job.
A lot of technical people think tech is the end, but no - if you get a small team to go knock on the person's door, and get your internet response team to shut down dns, or to get someone on site at the telco to perform certain actions at the router/switch level, etc all portions working together is a powerful way to resolve or to accomplish certain goals.
Think bigger, especially with state actors - the resources are there, and this line of thought is probably really basic stuff that people came up with in the 1960's or 70's (even when the arpanet was being created, there was probably already a team tasked with taking such actions - it only make sense to have 2 teams working on such goals in tandem - one to create the network, the other to take it down)