The embassies are not my ballgame but I get the impression they are not there because a lot of diplomacy needs to be done all the time - but to already exist when the need for diplomacy arises.

Open channels of communication and all that.

An alternate explanation to your point is that countries would like to have the appearance of being as shadowy and influential as what the CIA has going on. I'll quote (AFKA) mos def [0] and also link to the 99% invisible episode in number stations [1]. Some countries broadcast codes nonstop A in case they ever need to, no one can tell they just started and B to have the appearance they have covert agents behind enemy lines, just to be spooky.

  Mentioned that he worked for the embassy
  People seem to find that interesting
  High status, intrigue and mystery
  Special code name on the hotel registry
  I love it when they say
  "Enjoy your stay"
  They say it how they mean it
  'Cause that's the way they been trained
  Show you to your room
  A suite with a view
  When, if anything at all
  Do not hesitate to call
  ...
  Salutations, congratulations
  Reservations, exclusive arrangements
  Dinner with the patrons, the scenery is amazing
  It's so outrageous, they whisper when they say it
  When it's really real it's even realer than "The Matrix"
  Classic, modern, ancient, flagrant
  Get a special thrill every time they get to say it
  Peace! I work with the embassy
  On behalf of imagination industry
  I come visit, you come visit
  Such a pleasure, official business
  La, la, la, la...
  (Boogey, boogey, man, man)

[0] https://genius.com/Yasiin-bey-the-embassy-lyrics

[1] https://99percentinvisible.org/episode/numbers-stations/