You figured out the technique (which is essentially the same even if it's actually being sewn), it's simply Cutting Notches.
: puts on teacher hat:
IRON IT. When you assemble wrinkled fabrics, you lend yourself to tucks & sewn in wrinkles you don't want. Press that fabric before cutting it. Press after each pair of pieces are assembled. press the whole thing when it's done.
:removes teacher hat:
: puts on therapist hat:
Why are you so afraid of sewing? While your product turned out reasonably well, it will never be as good as a sewn one. What is the hang up?