The Klingons asked for Picard, what is the UFP to do? Say no and deal with a Klingon Civil War that could turn the empire against them or even just weaken their most powerful ally? Tell them they are sending someone else and they will just have to live with it? The Klingon response would be somewhere between go **** yourself and disruptor fire. They want Picard that is who they'll get.
As for Ambassadors sure the Federation has lots of them but not on every Starship, they are mobile moving between where they are needed be it other Federation Worlds or other Governments like Telarians or Gorn. It takes months to move around just Federation Space so this is not an on demand service we are talking about. Star Fleet officers are briefed on how to make First Contact but it is mostly a Hi how are you sort of deal. The Ambassadors only come in 15 minutes after the giant Crystal thing / Evil Computer / Deadly Virus is heroically torpedoed after one last plea for peaceful coexistence or Speeched at to either death or reform / threatened off by secret back up plan. By the time they get there the Enterprise has moved on to their next adventure.
Star Fleet Captains basically render aid when appropriate and make First Contacts, the diplomatic core handles all the boring stuff such as treaties and trade agreements. So no not really but Starfleet is usually the first on the ground anywhere so their Captains have to act like Ambassadors because what they say or do will have lasting repercussions for the Federation.