I am a recent convert to flatwounds, too. 21 years of playing for me.
I had tried flats for the first time about 10 years ago. I didn't like the higher string tension and less bright tone of the set I tried (I forget the brand now). Took 'em off and went back to rounds.
The second time I tried them was when I was studying some theory and picking technique from Carol Kaye's books a couple of years ago and corresponding by email with Carol. She recommended TI Flats, so I tried a set of those. Loved the feel and lower string tension, and the sound was good, but I ended up going back to both rounds and fingerstyle after a while when I got back in a rock band again.
Then, this fall, I joined a Modern Country band and have been experimenting with different sounds. I was considering the TI Flats again, but didn't want to lose the "edge" of my sound from rounds. In looking around, I discovered Darkglass Electronics pre-amps and picked up a B7K. With that pedal, I can dial up any kind of "edge" I want, it is amazing with tone shaping and distortion/edge, so that kinda freed me (psychologically if nothing else) to try flats again. I put the old TI Flats back on my bass, turned a few knobs on the B7K and nailed a sweeter tone than I've ever had before, a real (pleasant) surprise. So now the Flats are there to stay, I'm a total convert. For me, the biggest benefit is the feel and lower tension of the TI Flats, they are really something else. They are expensive as all get out, but totally worth it for me.
To answer the other questions, Modern Country and Classic Rock, and I play out 5-8 paid club gigs a month.