See, I think that out of the first four champions you listed, I think only one of them has an active reason to NOT join - Elise. Nami clearly joined to use her political power as a champion to expedite her search for the moonstone. It's not like she could very well search all of Valoran for one, so it makes sense to gain standing at the most public forum in the world. Syndra I feel would have joined the League to further expand her powers - what better training field is there? It's not like the Ionian government is going to spirit her away in the middle of a match. I can't see any specific reason for Zyra to join, but it's not like there's any reason for her NOT to join either.
Political standing has little to do with admission to the League, so I'm not sure why Draven and Fiora were brought up. They're both there for their own reasons, which seems to be a valid reason for champions to join up. We have quite a few champs who're only participating for personal gain. I can't see that the League has anything to gain by forcibly recruiting champions either - perhaps there is something that I haven't thought of, but in that situation why would they seek out somewhat dangerous beings like Syndra and Elise? That line of logic doesn't make a whole lot of sense, at least to me.
Perhaps there is some conspiracy going on in the League - the last JoJ pretty much came out and said that - but I don't think it has anything to do with the recent champions and their joining.
Syndra went bat**** crazy and murdered a man who was going to suppress her powers. She would most certainly not join the League who not only can inhibit her strength, but actively control her.
My info on Zyra is less concrete, but I was under the assumption that she was a very wrathful force that was just learning what it was like to be alive. Joining the League is a very odd move considering she would have no, or at least very little, knowledge of it. Maybe the problem here is how it was presented to us. If she were a topic of the JoJ for a while and then joined like Lee Sin it would make sense, but we saw it was she gained life and then immediately joined right after.
Nami I'm willing to relent. I've no idea how politically savvy she is. She may have some plan for all I know, but again this is probably another thing like Zyra where it just seems too sudden.
All that being said, the league has MANY champions that they've forced into cooperation. Even ignoring early incidents like Kayle, Morgana, and Taric, there's still Noc, Brand, the void monsters, and Fiddles sorta. The League has a lot to gain from these things. These incidents attract attention to the League and a possible pawn for them to use that no one would care about should they need to dispose of them.
Let's just assume I'm correct for a moment so I can give an example. Say there's some political unrest in certain areas, or there's an event that threatens to overthrow the league. A champion like Noc could "escape" and slaughter whoever is around. Since the process of summoning and controlling a champion has no visual queue, they simply send a batch of heroic champions to dispose of Noc and the League can play the whole thing off as a tragedy that they responded to with the best of their ability.