Product review (5 stars): The problem was...
... the lowest flame setting (as determined by mechanical detent, not judgment) on my gas stove was still so high as to rapidly boil items I was trying to braise.
The possible solution: a heat diffuser.
What I tried initially: a Fox Run diffuser as shown here on Amazon.
The procedure: a 10" wide, 3" deep Cuisinart stainless steel pan was filler with 1/2' of water and placed on the stove at the lowest burner setting; the time was then measured by stopwatch. After the results were obtained, the heat diffuser was inserted and the stopwatch re-started.

The results: after 6minutes without the heat diffuser, the water was at a low boil; after inserting the heat diffuser, the water diminished to a low simmer and remained so for 1 hour.

Conclusion: THE THING WORKS, and works very well. In my opinion, it is worth the money and performs as advertised. Long term performance still remains to be seen (i.e. will it warp or rust a month from now): suitability for long-term braising also remains to be seen. For stove-top braising, this seems to be just the ticket.

Possible problems: the wood handle became hot after one hour, but not too hot to hold; a very slight warp developed in the heat diffuser disc after one hour - but not enough to consider a defect; the handle and ferrule are so close to the disc that the pan can sit lopsided if placed dead-center (a possible fix is just to remove the handle - it's pretty well superfluous anyway - or just set the diffuser off-center and the pan dead-center).

Note: as a loose "control", I tried the same procedure with a Lodge cast iron Dutch oven; after 5 minutes, the water was at an extreme boil; using the heat diffuser remedied that; using a Lodge cast iron "trivet" as a makeshift heat diffuser had the same effect as the Fox Run diffuser when used with both the stainless steel pan and the cast iron pan.