No, I was serious. I could easily see the Federation only contacting the "major players" on any given planet. Perhaps only the technological and economic superpowers. On Earth today, depending on your definition of "a country," there are approximately two hundred countries, So while there might be some kind of courtesy call or polite chit-chat at a diplomatic conference, I really don't see the Contact Team interacting with every single country on a equal footing.
I think we need to make a division here between "Contact" and "Interaction".
I absolutely do not see the Federation allowing, to use Earth as the example, the P5 (the 5 permanent members of the UN Security Council) to be the only governments who know of the existence of extraterrestrial life. Even though the P5 might prefer that it be that way, not going to happen, that'd have bad impacts strategically and politically.
However, at the same time, it's impractical to deal with all 200+ governments at once, yes.
What I suspect would happen: All 200+ governments (in the Earth example) are invited to the initial meetings with the Contact Team. HOG (Heads of Government) and HOS (Heads of State) level only, no lower; No staff, no notes, no recordings. Such meetings tend to be zoos, even if you're dealing at that level, simply because of the sheer size, but that's to be expected, and it's fine at first. The point is that every government knows of the existence of extraterrestrial life; the decision on whether to reveal the existence of ET life beyond the governments' highest levels is (The Federation urges) taken on the basis of unaniminity - and if the decision is taken not to reveal it, the Federation won't say it (won't even hint
it), but they'd help to stop such revelations "out of school", and turn a blind eye as everybody else beats the attempted talker to a pulp geopolitically - after all, if you defy the will of the international community on this
issue, it may not be pretty
to see a country be ganged up on and basically torn to shreds, but it's evolution in action, because you are clearly Too Dumb To Live. (And yes, I believe that, while it would be difficult, you could do all that in an Earth-like society. I believe that the simple fact is that in such an event, while there would be a bias towards revealing it, it would be eminently possible to enforce a decision not to.) This would hold in case of Limited First Contact - when the Federation meets with governments on that planet, it meets with the highest levels only, and only all of them at once.
After The Announcement, though, once everybody on the planet is breathing again and the "Holy crap there's aliens! Let's riot!" riots (you know they'd happen) are done with, and after the initial impacts are clear (some governments would fall - some countries would cease to be independent too), then all the HOGs/HOSs (in the Earth example) of the still independent governments go back behind closed doors, and the Contact Team lays down the ground rules: Now that everybody knows which governments could survive the news going public, the governments of the planet are "invited" (read: politely demanded) to decide amongst themselves as to a usefully limited subset of governments that will make up the "Contact Group" on the local side. The Federation would prefer that this group contain a diversity of views, ethnicities, religions, etc, as represented by governments, but the idea is that it's a useful subgroup of governments. These governments get dealt with first - essentially, they're the conduit through which relations are conducted with the planet. How the governments are chosen is of no concern to the UFP - they know there'll be horsetrading, but that's life.
Similarly - every government gets a UFP consulate, just so the Federation has someone there to keep an eye on things, but only one capital gets the official UFP embassy.
Contact, thus, is established with every government. Interaction, on the other hand, is limited to a subset.