Passive perception just means that you don't roll, not that you are being passive. Once you understand how it is intended to work, there's no conflict to make sense of. You're already aware of anything that your passive perception total beats, and an active check is simply an opportunity to go beyond that.
Jeremy Crawford explains stealth and perception in detail in the Sage Advice section of this podcast, starting at about 15:09. He says that passive perception effectively acts as a floor for active perception checks.
Specifically, anything for which you might make an active Wisdom (Perception) check is something that you have already noticed with your Passive Perception, if that total is already high enough.
That means that the main part of your question is based on a misunderstanding. The answer to this:
Doesn't the Observant feat make you see more, when you're not actually looking? Or if you'd like me to rephrase that: doesn't it make you notice more, where you're not actively trying to notice anything?
... is no, the Observant feat doesn't only apply when you are not actually looking. It makes you notice more whenever you are conscious and aware of your surroundings.
So, there's no need to "make sense of" the opposite interpretation, the one that gives a weird conundrum. It's simply not a problem.
This means the feat is relatively powerful, because it can work in situations where you do have advantage, which adds another 5. That means that a character with +5 to Wisdom, the Observant feat, and advantage in a particular situation has a Passive Perception of 25. That's pretty crazy, but, hey, the player spent a feat for it. It's likely to be a significant part of that character's identity. Changing the feat to always give advantage would be a significant nerfing, and seems really unnecessary — after all, we're just talking about noticing things here.
From the podcast linked above:
JC (at 22:16): Now, going back to passive perception... this is, as its name implies,
passive. And, it's considered to be "always on", unless you're under the effect of a condition, like the unconcious condition that says
you're not aware of your surroundings. That really... the practical
effect of that is that basically your passive perception is shut
off. Passive perception is on basically whenever you are conscious and
JC (at 23:09): Because it's passive, the player does not get to say they use it. This is a... this is something that people...
Interviewer: (Laughs) I'm using my passive perception right now!
JC: Yeah, no. It's always on. That's the baseline. Now, this brings up questions, because then people are saying that, well, how
is it that when I make an active perception check, I might get a
roll that's lower? Well, you aren't... yes, that roll is lower, but
remember your passive perception is aways on. So it really
represents the floor of your perception.
Interviewer: Right. That's an important distinction, though.
JC: Yes. So if you make an active perception check and you get a number that's lower than your passive perception, all that means is
that you did a lousy job of this particular active search, but your
passive perception is still active. You're still going to notice
something that "blips" onto your passive perception radar. Really,
when you make that roll, you're really rolling to see "can I get a
higher number?" If you fail to, well, again, your passive perception score is still active. It is effectively creating that minimum.
Interviewer: The minimum. Yeah, I don't know if that's necessarily clear to a lot of dungeon masters out there, because they will be
like, well, the opposed nature of this roll means that you were
just really bad at looking, and even though the person who is
sneaking up on you only got like a five, they're able to do so.
JC: Now, many of these sorts of situations would be erased if DMs just simply remembered to use the passive perception in the
first place. Because honestly, if something's noticable by a
person's passive perception score, they should already have noticed
it. So really, the active search is trying to find something
that you haven't already noticed,
and your passive perception score represents what you have already noticed.
(Bold added to highlight the key points; italics intended to represent emphasis in the speech.)