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.

[0] https://en.wikipedia.org/wiki/Pantheon,_Rome

[1] https://en.wikipedia.org/wiki/Harry_S_Truman_Building