Wasn't it just a coalition that would eventually become the Federation?
Genuine question, as I've not watched Enterprise in 15 years.
It's an oddly worded scene.
(Riker joins Troi at the back.)
TROI: Is he nervous?
RIKER: Wouldn't you be?
TROI: Oh, he'll be fine. I had to memorise this speech in grammar school. You wish you could tell them all that this alliance will give birth to the Federation.
RIKER: I think I'm ready to talk to Captain Picard. I should have done it a long time ago.
TROI: So, I guess we're through here.
RIKER: I guess we are. Computer, end programme.
I take that as being a bit less literal. I don't Troi is saying "Hey, don't you wish you could tell these people here at the founding ceremony of the Federation and this will give birth to the Federation?", but rather speaking a bit more abstractly that what they are doing right there, at that moment, will give birth the Federation as it is in Troi's time, what it becomes.
It's also possible that it's not "The United Federation of Planets" just yet. They have the logo and stuff, but there may have been an interim proposal somewhere in between the Coalition and the Federation that was short lived.
As for the DSC stuff, i'm ok with them going on a TNG's greatest hits tour. It's... a weird setting for it, but whatever. I have appreciated that the longer DSC goes, the less inclined they are to "visually update" everything, or at least in drastic ways. The Romulan Scout Ship was perfect. It wasn't EXACTLY the TNG ship, but it was the
shape of the TNG ship just with some more details on it. And 24th century Romulans looked like... 24th century Romulans.