I don't know how helpful of me this is to contribute, but I know an ENFJ - Enneagram 3, which doesn't help - who often seems essentially perfect. Part of the problem may be that as ENFP, I appreciate E, N and F, and because my P makes it hard to function in the world, I wish for a J switch - so of course the "perfect" solution to that would be ENFJ. I would imagine FPs would tend to idolize ENFJs more than other types.
And please don't take this post wrong, it's really quite a compliment. I usually love ENFJs and I don't mean to generalize for everyone. I'm just thinking in terms of the ones I know.
In general, I find that the ENFJs I know - 4, maybe 5 for certain - are friendly and enthusiastic but not melodramatic, you're outgoing but still private, you are empathetic but not overly emotional, you're smart but you have "street sense", you're confident but not cocky, and you're generally quite successful and still humanitarian. The ENFJs I know also do tend to hide their problems, which means that generally most others never get to see the weaker, more vulnerable side of them. You guys generally project confidence. Even when I
know an ENFJ well it's sometimes hard to remember that you guys struggle inside a lot too. You seem happy, successful, creative, compassionate, and in control. Capable and self-assured. (Haha wow that was a lot of c's

)
And seriously, even ENFJ complaints sometimes sound "perfect". Of course it's repulsive when someone has no concerns at all, but ENFJ concerns often sound like just the right type to make you like them, lol. For example, that friend of mine was just worried about getting into a plum of a job that she was selected out of a huge group for. Sometimes she's worried about whether the people she looks up to will like her. She's stressed because she cares too much about other people's problems. She doesn't take enough time for herself. She has a lot of responsibility on her plate because she gets in really high-level positions. I mean look at your post - you're
concerned about someone finding you perfect. I would be privately reveling in sheer delight if anyone ever said that to me. I totally know that you're not, but if I didn't know other ENFJs better, it would almost seem like you're trying to brag, lol.
I don't say this to make anyone frustrated, at all, but to maybe help ENFJs understand why it seems like that to some others? It's particularly true because you guys are generally quiet about your really deep dark down problems, and so outwardly happy and well rounded and successful. Even when you express your frustrations, many of you do so in a way that makes you sound really good. From an outside perspective, it's like... why are you complaining? I'd be toasting myself every day if I were you.
The problem is I think the antidote to this perception
is to share your really vulnerable places, to not close off from others when you're frustrated or scared. To be okay with showing yourself as really, truly weak and flawed, not imperfect in a good light. To just... let go. Drop the wall between you and the other person and just trust them to be able to handle your darkness. I know that maybe it feels like you're "dumping" on the other person, but I am relieved when someone opens up like that with me. It's so much more of a comfortable state of presence for me. Maybe some of you already do that, probably a lot of you do it with just a select person or two. It's kind of liberating, if stressful.
It's sort of up to you if you actually want to make that tradeoff, though. I've opened up way too much my whole life, so I don't really have any hope of appearing perfect. But truth be told, if I were in that position, I might not want to open up either.