This is good info, but I think Highbeam or BKVP can give you better answers than me, as I've never seen any behavior like this from either of my BK's. If you had said 11 o'clock is the new 2 o'clock, I would've just told you to mark that new position on your dial with some tape, and use some position relative to that to set the stove for cruising in the future. But the fact that turning it to 11 and then right back to 2 gets it cruising as desired, it would appear that indeed something must be sticking inside, which I've never seen on my stoves.
The mechanism is beautifully conceived, in it's outward simplicity. Your control knob drives worm gear to translate shaft direction, and this drives an axle around which a bimetallic coil is wrapped. Bimetallic coil looks like a clock chime coil (shown below), with your knob driving the center and the air flapper connected to the outer end. The whole "spring" turns with your knob, moving the air flapper with it:
View attachment 304207
As this spring heats and cools, it pulls or pushes the flapper a
little more open, or a
little more closed. But as small movements in an air control translate to big changes in burn rate on any stove, the movement is small relative to your turning of the knob.
Getting the ratio of temperature to degrees of movement, as well as the critical time constants required to prevent oscillation, is where the beauty and real engineering comes into the design. Several other stove companies have tried similar mechanisms with various degrees of success or failure, but BK is one of the few that really nailed this.