Ordinary individuals who attempt to use technology to 'go dark' will likely, if they're careful, be able to conceal their online identity from your average stalker type, malicious ex, or seedy credit card fraudster, but certainly not from the likes of Google, Apple, Verizon, or any number of government agencies with access to those outfit's data centers. This is called mass domestic warrantless surveillance and it is illegal under any reasonable interpretation of the US constitution, and certainly shouldn't be allowed by private parties either.
As far as spies, every history of spies I've ever read has one thing in common: they all hide in plain sight, using cover identities of some sort, acting as much like a normal member of society or their organization as possible, and then running off to do their data transfer/nefarious activity in secret only rarely. Notably, mass domestic surveillance isn't a very useful tool for catching such people - it's more about authoritarian snooping on the population, engendering fear of the state as with STASI, as a means of control. That nonsense shouldn't be allowed, and those who promote it are nothing but authoritarian enemies of democratic rule and free expression.