It's complicated, but that's kinda what happened to the Lavabit "secure" webmail service. When the owner wouldn't install a backdoor, the FBI sought the private encryption keys so they could MITM the whole site.
The big Apple vs FBI high profile lawsuit for one. Granted that was a telegraphed precedent seeking exercise by "accidentally" losing access to the work phone after the terrorists destroyed their personal phones before the attack.
True - they are less publicly tested but it is suggestive in its own way. If they could just NSL their way to access why bother with precedents? It is wild speculation but that is what lack of transparency has wrought.
Edit: I may be thinking of a FISA order as opposed to an NSL. Doesn’t matter though, obviously the concern is that they would be served with whichever does allow that.