The 5557 works well, too (Just a CM7 up 5 half steps).
John, I'm not fully sure how to respond to your question, and I've had a lot of theory (I hold a PhD in music).
There are two things at play:
1) What chord you're playing on your ukulele
2) What the song actually does
As I make the ukulele play alongs, much of the music is pop music, and without being snobby about it, it's pretty straightforward and non-complex. But that's why pop music is so popular--we understand it, it moves us, and we can understand it. And with some musical experience, we can even play it.
Now, are there more complex and "heady' pop songs? Sure. But they generally don't hit the Top 40, and they don't make as much money.
Other genres (including jazz) tend to be more complex, harder to process, and require a greater variety of chords than the "easily consumable" pop music.
[At this point, if I sound like I am being negative about pop, I'm not at all, but we're talking theory]
When it comes to music, we have an existing syntax of musical language where, in a key, certain chords function in specific ways. And the syntax is so built into what we hear (even if we don't know theory) that if a song doesn't follow those conventions, it is shocking to us.
You mentioned this progression: F Am F7 Bb C
In the key of F, the chord functions are: 1 (I/tonic) F, Am (iii/mediant), F7 (V7/IV-dominant 7th of the subdominant), Bb (IV/subdominant), and C (dominant).
Chords generally flow in this structure: iii -> VI -> IV or ii = > V(7) or vii -> I
Your chord progression follows this, skipping the submediant (vi) or dm chord, but instead shifts over to borrow a chord from the key of Bb (the V7 of Bb, or the F7), and then jumps back to the sequence. I'd be very surprised if a C7 didn't work well at the end of the progression instead of a C, as the C7 would lead more fully back home to the F chord.
So, the Am makes sense from a theory point of view.
One of the great things about the ukulele is that chords are much simplified with a maximum possibility of 4 notes. In music, we spend a lot of time with triads, where there are three chord tones...and then for voice (four parts) we double one of the notes. There are rules about voice leading and so on, but that's the basic idea.
Then we add seventh chords (minor, major, half-diminished, and fully diminished) which serve other roles.
It's only when we start going beyond the four note chords to 9th, 11th, 13th, etc. that the ukulele has to make substitutions.
One of the things that players do, and this is mentioned above, is that they play PARTS of other chords (that might have 5 notes or 8 notes) by playing notes contained in that chord, but not the whole chord.
A simplified example of this is the Hawaiian D7 2020 (ACF#A), but we call it D7 most of the time, as the D is covered by the voice or the bass (if you have one). Realistically, if you want a D7, you would play 2223 (ADF#C), but we're not too picky about it.
And then the other aspect of music that (intentionally) forget with ukulele is the idea of chord inversions...which note of the chord sounds the lowest when played. In traditional music, the lowest sounding note impacts how the chord should be used. A C chord (CEG) where the C is the lowest sounding note is used differently in a harmonic progression than a C chord where the E is the lowest sounding note. Ultimately, a chord in "root position" where the lowest sounding note is the name of the chord (e.g. C on a Chord) is the most stable...and even more so when the highest sounding note is a C as well.
Want to know why the C chord sounds so good on reentrant ukulele? The C is the lowest sounding note, and the C is the highest sounding note, all at the same time. (0003 GCEC)
Theory is pretty fun--but most people know what music is supposed to do, even if they cannot do it themselves, because we are surrounded by this music syntax before birth.
I know that went on too long, and I'm sorry, I was just enjoying thinking about it. As I teach elementary school (ages 4-11) I don't get many opportunities to stretch all that learning that I did in the past!