With the disclaimer that this is the result of my reading, having spoken with a musically trained person who heard a live performance and having listened to some online mp3 recordings, here's my explanation:
Each part in Sacred Harp(treble, alto, tenor, bass) has a bit of harmony that weaves in and out of the melody. Sometimes, only the alto's will have the melody, but in many cases, the melody 'skips' from one part to another -- just as you said your improvised treble sometimes was lower than the soprano. I do this also with my voice as I improvise a higher harmony that often mimics the tenor, but sometimes does not (especially if there is a song without the tenor). If the soprano part is too high in places for my voice to go above, I drop back to the melody for a few notes or harmonize a little below it, returning to the higher harmony when I am able.
As I understand (again from reading), Sacred Harp is designed for a participatory experience. In other words, there isn't really anyplace for an 'audience' to sit and listen. Non-singers often sit behind the tenors to listen. If you picture the points on a compass, the singers form a square facing inward. The tenors are at North. The altos are at South, trebles are West and Basses are East. A director stands in the center and offers the pitch. Each voice part gauges how many steps above or below the pitch they are by the shape notes on the page. Using the shape notes, they sing one round voicing only the shape notes (not the lyrics). After this, they begin the same song with the lyrics and singing their respective parts. In this way, the participants are singing notes relative to each other and not necessarily a perfect reproduction of particular notes of written music.
The reason it is considered participatory is because the best place to stand and hear all of the parts together is in the center of the square (where the director stands). The next best place would be to stand and hear the singing is as you participate in the the music.
It's interesting that you started doing this using a flute. I would think that this would work better with wind and string instruments than with others -- although I cannot quite imagine how it would sound.
A thought that occurred to me is that when we sing in 'that city built four square' I wonder what kind of vocal instruments God will grant us in our new bodies and if there will be songs of praise that blend and ring like this as they wash over God on His throne (in the center) and splash back over us? What an unfathomable time!

I've found two sites that you can listen to to 'hear' this type of singing. This first includes interviews with some of the participants. I don't necessarily agree with the one person that calls is 'not necessarily pretty' and the one that calls it 'shout it out harmony' because I think that the director can soften and raise parts to create a pretty and resounding performance without it being so much shouting

.
I particularly like the renditions that have fewer participants such that you can hear and get a better feel for the harmonies as in "The Promised Land."
http://www.npr.org/templates/story/story.php?storyId=1534280I also like how one participant describes the experience:
"The room vibrates with sound. It rises up into your feet, pulses through the floor. You can feel it buzzing through the tunebook in your hands," [Melissa] Block [reporter for NPR] says.
Singer Buell Cobb says the power of the gathered voices is so strong, it feels like a solid force. "It almost seemed you could stand up and walk on it."
The other site is of large Sacred Harp conventions. Again, you get the idea of how the music goes, but I think you'd have to be there to really appreciate the experience of the sound. My favorite on this site (today

) is 'The New Jerusalem' composed by Jeremiah Ingals.
http://www.pilgrimproduction.org/sacredharp/carrollton/chatahoocheeharp.htmlRemember that as you listen to these, the first round is just a bunch of notes, the lyrics kick in on the second 'go-round.'