Literally every single muscle in your body is, or at least supposed to be responsible for maintaining good stability. The body has lots of different small stabilizer muscles employed for this very purpose, but larger muscle groups are also working, such as your lats for example.

Your overall core strength as well as your lower back are both active in maintaining stability and work closely together . This is why people with lower back pain still deal with issues in that area even after training their spinal erectors. Training your abs will actually help tighten your backs posture more or less into a more suitable position where the rest of the various muscles in your back can now keep everything tight and in a more reasonable posture. This will certainly help contribute to power delivery.

There is no "option" here, whether you want to use specific muscle groups or not. If you're riding a bike, your whole body is responsible for both stabilizing and transferring power- period. Think of the energy transferring from your body to the bike as being one. You want good direct power and steering inputs to your bike without being overly stiff.

Being forced to stay overly mindful of your posture and which muscles are contributing to stabilization as you ride is obviously not a good thing. But id still rather do that than ride down the road like mr. Arched banana back x floppy noodles.