You'd have to go REALLY back to basics... a lot of chipsets have backdoors themselves.. some of which the FBI ironically pushed for. So no matter how securely you make your own OS if you don't also make your own hardware it's all for naught.

Also if the government started making its own computers then private suppliers would raise a fuss about "big government" and hurting private enterprise. Also can you imagine the shitstorm ensuing from: "Government insists every tech company bakes in backdoors. Government then has to create its own computers because consumer models are too insecure."