Today’s song with basically perfect chords:

Whitney Houston – Saving All My Love For You

I can’t remember if this amount of jazziness bothers you; for me it’s a reasonable amount. I hear the verse chords as I → vi → ii → V:

A [A]few stolen [F#m]moments is [Bm]all that we [E]share
[A]You've got your [F#m]family and [Bm]they need you [E]there. 

…with some 7s, 9s, 11s, whatever.

The pre-chorus throws in a couple nice vi → II changes:

Though I [F#m7]try to res[B7]ist being 
[F#m7]last on your [B7]list,
But [A]no other [F#m7]man's gonna [G#m7]do [C#7b5]

…where clearly the III7b5 at the end is the best chord in the song.

The chorus takes us back to a familiar IV → iii → ii → V → I:

So I'm [Dmaj7]saving [C#m7]all my [Bm7]love [E]for [Amaj7]you

The bridge is pretty great too:

[G#m7]You used to [C#]tell me, we'd [F#m7]run away together
[Bm7]Love gives you the [E]right to be [Amaj7]free.
You [D#7]said be [G#]patient, just [C#maj7]wait a little longer
But [Bm7]that's just an old fanta[E]sy

If you’re keeping track of non-diatonics, that’s two III’s, a #IV, and a a VII.

Summary of quality things that happen:

1. I never object to a vi → II.
2. That III7b5 going into the chorus. None of the transcriptions I can find online even mention the b5 in this chord, which makes me sad. I also don’t quite know whether to call it a b5 or a #11 or what the difference is.

I think when I say "jazz" (in a derogatory tone), I mean "chaotic randomness", like prog-jazz more than pop-jazz. In fact this song is jazzy in a way that I think is awesome that you didn't even get to in your list of awesome things: everything I like about good melodies is packed wall-to-wall in this song.

For example... I agree, the verse is a standard I → vi → ii → V. I like this transcription, which calls it Amaj7 → F#m7 → Bm7 → E11. In any case the bass is very clear, I → vi → ii → V.

What do you think the melody is doing here? Following the root? The third? Not so fast. Here's the chords and lyrics again, with melody notes in red:

A [A](G#)few stolen [F#m](E)moments is [Bm](C#)all that we [E](A)share

Not crazy to the point of non-diatonic, but none of the four main notes here are in the a triad they're sitting on top of. None! If you're keeping score at home, this melody sings (relative to the chords) a major seventh, a minor seventh, a 2nd, and a 4th. This requires an insanely good singer to pull off... and I'm always impressed by this sort of melody, in the sense that in my limited songwriting experience, even if I don't do it on purpose, what I basically do is ask "okay, which chord tone should I sing here?". Maybe I'll toss in a sus4 once in a while. But how does one even think of singing G# E C# A over A → F#m → Bm → E?

And while I'm only sort of susceptible to the allure of "big notes" (I am, a little), I'm totally rung up by big notes that come at the top of a big arpeggiated melody line with big intervals in it, as in "no other man's..." (C#4 E4 F#4 C#5).

Then after that is your favorite chord, which is also my favorite spot in the song, but not because of the b5 or lack thereof (honestly, I don't hear it... can we still be friends?). I love it because it's definitely a III7, and the melody is still holding a confident dominant 7th (a B over a C# chord). Have I mentioned (I ask rhetorically, because I know I have, a zillion times) that I love dominant 7's in the melody?

But wait, it gets crazier. Here are the chords to what you called the "chorus":

So I'm [Dmaj7]saving [C#m7]all my [Bm7]love for [Amaj7]you.

Also FWIW, I actually prefer the Glee version, which is faithful to the chords but adds (1) just the right amount of harmony, which for me is never zero, and (2) just the right amount of "a dude singing high enough to strain a little", which for me is also never zero:

Glee Folks – Saving All My Love For You

I only watched the first season of Glee, so I have no idea what's happening here. Without context, the scene around 1:40 that might be flirtation or might be physical therapy or might be child delivery is really uncomfortable. Though not nearly as uncomfortable as the hippie dude singing "Saving All My Love For You", and singing it well.

I am going to raise one criticism of this song. Either of the following would have sounded fine and made grammatical sense:

1. 1. I tried to resist being last on your list, but no other man's gonna do.
2. 2. Though I tried to resist being last on your list, no other man's gonna do.

What she actually says doesn't make grammatical sense, and doesn't add any aesthetic value to the rhythm IMO:

3. Though I tried to resist being last on your list, but no other man's gonna do.

"But" really wants to be preceded by an independent clause, and "though" turns the pre-comma portion into a dependent phrase. I don't think music needs correct grammar, but for some reason, this one drives me crazy. SO FUCK THE FUCKING '80s AND THEIR FUCKING DEPENDENT PHRASES.

Also another point on the Glee version. If the YouTube comments are to be believed, and if typical Glee casting standards are to be believed, the dude with the dreads is profoundly good-looking. If this is true, he rockets to the top of my list of "guys who are apparently profoundly good-looking whom I never would have guessed are profoundly good-looking". Not that he looks odd or anything, I just would have had him down as "dude with no obvious physical flaws".

A confession: I still can't tell that Brad Pitt is good-looking, even though he is clearly the media's archetype of "good-looking". I can only tell when guys are good-looking if they have giant shoulders and ripped abs, like Matthew McConaughey or the guy from "Arrow". Another confession: I watch "Arrow".

1. I do not prefer the Glee version. They took out my b5 (or #11, whatever you want to call it) from the prechorus, which I swear is there in the original, and they took out the bridge. No amount of male dread-headed hotness is enough to make up for such removal of interesting musical features.
2. The dread-headed male looks like a guy I knew in college who was indeed considered attractive by most females. That's the only method I know for deciding such matters, comparison to exemplars.
3. Sometimes bad song grammar doesn't bother me, sometimes it does. This one doesn't. The one that does, more than any other, is "what if god was one of us".