Actually, I'd argue Mozilla wouldn't want a "best viewed in Firefox" campaign since the standards should apply to everyone, even if some browsers do better with some feature sets than others. There are already existing ways to show your support, such as https://mozilla.github.io/for-firefox/
I'm a Mozilla employee. I'm not speaking for the org as a whole, but I share your view, and I'd expect most/many other employees to agree with you on that point.
Pages should strive to be compatible with all browsers, and self-dealing in the space of compatibility is something that would go against the ethos and mission (IMHO, and I would expect many others).
That said, I expect that we would support a similar campaign, just advocating for Firefox using other points instead of "best viewed" style compatibility.
Again do not read too much into my specific choice of words. It could not worded that "Use the best standard adherent browser like FF" or similar.
If Mozilla's mission is a more open and standard compliant web instead of walled gardens that message needs to be conveyed to ordinary users and explained as to why it is better. I am not a copywriter so I do not what that messages should be but I think it should be done.
Of course - I'm not assuming anything much about your suggestion. It seems like a great idea in fact.
I personally don't have a read on how much support there would be among well known independent/commnity-aligned services for such a campaign. If there did exist a widespread sentiment of that sort, it'd be great.
