The timeline is plausible (even if the premise is questionable).
I don't see any catastrophic issues with the details of your timeline as you've laid it out. So in short, my answer to your question is yes.
I'm sure you'd like some more feedback than that, though. So first, here's a few points I would like to address about your scenario:
"Neither Nixon nor Bush 41 would feel it's worth censoring." Politicians will do whatever benefits them politically. If the fallout from censoring is less than the fallout from not censoring, they will choose the former if it's within their power.
The government already has tools to to examine Internet data, and it would be possible to pass laws to further ease their ability to do so. For example, requiring network hardware manufactures to include software allowing government back door access into systems (which is obviously a potential weakness that a third party could exploit). Such technology already exists so it's demonstrably possible. Enforcing the use of "government approved" network hardware could involve validation of this software's existence in internet requests. It would still be possible to fool, but it's a potential element to consider.
But that doesn't stop sources outside the US from releasing the information. The government would still block anything it didn't like regardless of it's point of origin, but there's nothing stopping Snowden from bringing a flash drive to a WikiLeaks-esque organization based outside of the US for the rest of the world to see. And once something's out of the bottle, it's hard to put it back in.
- The 2016 election outcome: Tough to say. For starters, you can be sure that essentially any and all of the negative press Sen. Clinton received for her ill-advised communication practices would have been quashed. Speaking generally, I would incline toward a left-leaning press having an even easier time propping up liberal candidates and negatively portraying conservative candidates (that is, objectively speaking, the majority of news media tends to lean toward the liberal side of things, and I'm assuming the political climate in your scenario is comparable to our present day environment). So, if you would prefer a Clinton victory in your story, it would be believable, though I still see it being a tight race. Personally, I would think a Trump victory was still likely; without any smoking guns from some particularly shady hypothetical dealings by the Trump campaign, I feel the US is too polarized at this point for almost anything to sway too many people one way or the other, but serious enough matters could still affect turnout.
Now, the premise itself of more lax prior restraint restrictions is problematic. Something like a News Management Act is inherently against the spirit of the First Amendment. It's not an accident that this is the first amendment in the Bill of Rights. Don't underestimate the importance of protecting speech in America, throughout its history.
I would suggest that in order to make this scenario more believable, you devise some new occurrence (or alter the details of New York Times v. US) to create a situation not directly related to national security, yet with a strong possibility of an indirect implication that is difficult to prove with certainty, yet significant enough to lead the court to decide to play it safe. This might be a good explanation for an alternative ruling to the 1971 case or something like it.
All in all, this is a very thoughtful scenario, and certainly a plausible one for a work of fiction. I encourage you to continue with it. Well done!