Yea, I been thinking about that one. I have to wonder if the 2 and 3 can even share the same world since "Different company". So any logic from 2 may well be invalid because of that. Kinda like how there's comics compared to films. There's certainty a fair few in 3. But damn is it fun when they fly in troops.
And this is proven when? Even if a president goes "We're the only ones around" it could be in the context of "On this base" or "In this part of the country with the NCR". And Eden may well be left out of the loop with the faction on the whole and a puppet that is used. And to be honest, he does strike me as the type to be used in such a way. Give someone power. Let them enjoy what they have while masking the rest of it. Games have done such things before.
Also, keep in mind that games can ditch logic in favor of fun. Sometimes things get added in for the sake of it. Deathclaws are all around for example. Do they have an explanation for being all over? Do they even need one? I see no Geckos. I wanted geckos. deathclaws here and geckos not? Where's da logic?
You're free to speak for yourself and say "I personally think they should have been gone", others can and will disagree with you. Just like others can and will disagree with me. There's no should about it one way or the other. I'll give you the benefit of the doubt and assume you was speaking only for yourself and those that have a personal preference against them.
Pretty sure you got recruited in 3. And while the Enclave detest outsides, they can and will use them. Keeps their more pure members safe. Which, for all we know, may be sipping G&Ts on a none radiated beach. Not every pure human is bound to be a soldier. And again, who knows if they have cloning technology or not. I can't recall if Fallout 4 has done clones yet. I do however know other games have used this tactic. Both on small numbers and large. And even the Gen 3 synths are grown in tanks, so it's not that large of a gap between the two. Enclave have different technology to the Institute of course, but we're still talking "high tech" and it's not impossible for similar tech to be made, despite not knowing of each others presence. Institute may well be remnants of the Enclave that used to believe in the what the Enclave did and now no longer do, since they share VERY similar beliefs in terms of "outsiders". If that's the case we pretty much just finished them in 4.
You know, that last explanation probably makes the most sense. Lose bases on surface. Hide away underground. change beliefs over time. DLC with that confirmed is a nice compromise. And Shaun could be teleported out the moment the place blew to confirm it. While still clinging onto life and then dying before your eyes without hating each other. Wouldn't even matter if the base got blown up or not. 'Cause if it was it wouldn't be Enclave anymore anyway.