It's hard to explain, but serif typefaces have centuries of history behind them, with practically all the books (in Europe and America) using some variant of serif since basically the invention of movable type. You can add most of newspapers to this list (notably "Times" part comes from a font being cast for the British newspaper The Times in 1930s.) The Roman Empire is a source of these (just look at the inscription on the Pantheon[0]), so that's for the "Roman" part. Using serif fonts signifies "tradition". Using sans-serif signifies "modernity". Being "stately" is more about the former than the latter.
Interestingly, the building that houses the State Dept[1] has its style defined as "Modern Movement" and "Stripped classical", which is interesting, because it's somewhere in the middle between modern and classical, while not being quite any of them. To me, it looks mostly bland; I'd much rather have an office in Congress building, or Supreme Court building, or the White House, all of which are strictly classical.
Not going to argue about personal interpretations, and everyone's entitled to have one — including people who have interpretations that we personally disagree with.
But to take an extreme example: at least at first glance, how would treat a SCOTUS decision written in Times New Roman versus one written in Comic Sans?
Serif fonts in general are more formal (or "stately"). That's because of the, well, serifs. Sans-serifs visually resemble/evoke handwriting because in handwriting we don't make serifs. Try writing with serifs for everyday stuff and it will quickly reveal to be visually and logistically annoying.
The only question is which brand of formal do you want to evoke. TNR feels very straightforward and no-nonsense to me. So I wouldn't really object to associating it with bureaucratic memos and documents. For something more leisurely I'd go with either Garamond or Georgia. Georgia is my favorite but it's a bit too fancy for long texts. It's better suited for quotes.
As an aside, I hate that Introduction to Algorithms [CORMEN] uses a TNR-like font. It makes it look like a Physics textbook. They could've stayed with default Computer Modern and strengthened my Pavlovian conditioning to it. Whenever I read something in Computer Modern, my heart palpitates a little in the way it did when I sat through my Data Structures exams.
Georgia is nice, but rather recent. Garamond, Baskerville, Bodoni, Caslon, Century are all centuries old. If we're talking about the US State department, it would probably be fitting to look at the fonts that were used for US printing and communication (and are still used.)
