Chris, I'm not stating it's just the weight - It's of course much more complex than that!
And most likely it's also a combination of different little details that make a difference,,, MY advice is; Relax and let the Nikon techs do their thing.
Interesting observations. Obviously, it is Nikon's responsibility to solve this problem(s), but i suggest it will be enlightening run off coordinated empirical tests. Although the logistics appear challenging to get together enough copies of the offending lens in this case and say 4-5 bodies (Df, D750, D800, D810 and DX's etc). The ideal would be to get together several 300 f4 lenses in the 2 categories (i.e well behaved/ no issues Versus Shaky) and test them on the same standardized images (this test sheet can be circulated electronically). With standard settings (e.g 1/30 >>> 1/1000) with VR On vs Off). Cameras with mirrors up vs usual function, and tripod collar on vs off, and of course on different DX and FX bodies.
The internet can take this process a long way forward. Many of us are positioned to run off such tests using a std test image that we can pool then the images to a willing analyst.....Suitably one or two Nikon owners who have bought a troublesome copy of this 300 PF can then summarize and bring the results to Nikon's attention.
At the moment I don't own a 300 f4 PF [been put off so far!] and only have a D7200 - Df scheduled for later this year :-) but in my case, I can ask around in the Cape for willing collaborators. Indeed I know at least willing 3 friends, and we could arrange to test together with a D7100 and D5100 etc.
Basically this is doing heavy lifting on behalf of Nikon, but purely a suggestion. Where some Clients have been waiting for too long - and there is yet to be informative Feedback from Nikon - such a coordinated test will provide fairly statistically solid empirical evidence....