Each time you successfully complete a touch action after triggering inner quiet, inner quiet grows. By itself inner quiet is useless. however, by simply completing any touch, you get a quality bonus from inner quiet for each action you complete. If I remember correctly, Byregots gives a 20% bonus per inner quiet, starting at 100% quality. So, with 5 inner quiets (5 successful touches), you get 200% quality for a successful Byregots. If you can make it to 10 inner quiets you would get 300% quality.
As far as I can tell, you do not get the quality bonus that inner quiet gives you, so, it takes about 6-8 successful inner quiets before Byregots becomes better than Advanced touch (and it does cost 1/2 as much as advanced touch in CP). So I typically save up for one Great Strides and an advanced touch followed by a Great Strides and Byregots to get the biggest bang for my CP.
Inner Quiet by itself is far from useless. Doing a 1 star recipe earlier I didn't use Ingenuity, only Hasty/Basic Touch. First one to land was 105, 11th one to land was 279.
Byregot's Blessing does count in the +20% per step quality bonus, as well as the +20% per step efficiency.
Innovation doesn't work as well as it sounds; you get +50% (one place in game says +30%, but it's +50%) bonus to your base
control. So 5 steps of Inner Quiet and you are at 200% control. Use Innovation and it's 250% control, not 300%. It's still 2.5 steps of Inner Quiet for 3 moves, but another slight disappointment after I hit 50 in a craft.