Didn't "Yesteryear" establish ShiKahr as the capital city of Vulcan?
No, just as the city where Spock's family lived, nothing more. There's nothing in canon suggesting it's the capital, although the script for ENT: "The Forge" makes it the site of the Earth embassy. I believe it was Spock's World by Diane Duane that portrayed ShiKahr as the capital, although J. M. Dillard dissented and established a city called ShanaiKahr as the Vulcan capital. Jean Lorrah's The Vulcan Academy Murders made it the site of the Vulcan Science Academy, but I don't think it was said to be the capital there.
Fair enough. And in the age of interplanetary states, there is of course no particular reason to assume that the embassies of foreign states must necessarily be located in the capital city, though that is the common practice amongst modern states.
Who says there's only one capital city in the strict sense? Any number of Earth nations have multiple capitals.
South Africa has three capital cities: Cape Town, as the seat of Parliament, is the legislative capital; Pretoria, as the seat of the President and Cabinet, is the administrative capital; and Bloemfontein, as the seat of the judicial system. In the Netherlands, Amsterdam is the capital city but the parliament and courts are in The Hague. And so on.