Mike,
You're exactly right about the circle of 5ths helping you with progressions. For instance, if you're in the Key of "C" and you're doing a 2-5-1 progression. First look at the "C" scale
C D E F G A B
1 2 3 4 5 6 7
The 2-5-1 would be: D G C
Now look at the circle of 5ths.
C F Bb Eb Ab Db Gb B E A D G
If you start on the D chord, the next chord is G, then you're back to C!
Please note that I don't want to confuse the issue but the D-chord in the key of C is probably a minor but that's a whole other topic. However, this would be the reason to learn major, minor, diminished chords etc in every key and in all inversions.
I rarely think about the numbers. I'm not saying that's a good thing, I probably need to study that more.(Especially when a music teacher starts talking about "the five of five". Again another topic.) But I think it's because I learned the circle first and it works. I do remember the first time the circle clicked for me! I was playing a congregational song in church in the key of "F". Somehow I ended up on a D-chord, just without thinking I went to a G-chord, then to a C-chord, then I was back at F. I just KNEW to do that because I'd studied the circle. If you look at the numbers, the Key of F is:
F G A Bb C D E
1 2 3 4 5 6 7
So what I did, almost without thinking, was a 6-2-5-1 progression
Here's the circle again
C F Bb Eb Ab Db Gb B E A D G