On old branches like that, I usually place grafts on top of the stock branch, rather than on the sides.
And yes, I cross the scion across the stock cambium. Not at as much of an angle as you showed, but by crossing, you know there’s a point of contact. It’s really hard to position the cambium layers so that they exactly line up straight.
To ease the problem of the flaps breaking off, make a shallow slice on the stock, then move your knife back, and make another slice, then move it back a bit more and make the final, deeper cut where you’re going to place the scion. That eases the pressure a bit, and makes it easier to insert the scion.