Step 3Cut off mortices on sticks and shelves
In the sticks make a mortice for every shelf.
In the shelves make a mortice for every stick.
In our case this means 6 mortices in each stick and 4 mortices in each shelf.
As a rule, in sticks make a mortice 2/3 deep of the side of the stick and in shelves make a mortice 1/3 deep of the side of the stick. (In the example this means 4 cm in sticks and 2 cm in shelves because the side of the stick is 6 cm).
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I am sorry but it is not a good idea. The whole desing is based on structural simplicity. And puting screws to hold the selves to the sticks would destroy this simplicity. Furthermore, mortices + wedges is much much stronger than screws.
But yes, eliminating the need to wedge would be desirable. Wedging is rocket science for a clumsy dude like me.
Unfortunately the desing does not allow to eliminate the need of mortices (which is the most dificult part of this construction). And the wedges are there because of the mortices. In a platonic world wedges wouldn't be needed at all because mortices and surfaces would be perfect (a.k.a platonic). Wedges are introduced because we all are human and then imperfect (as our constructions are), and wedges is a very efficient way (after Archimedes) of filling up gaps; (as when we put a folded paper under a table leg).
On the other hand I do not give credit to your statement of not being able to deal with wedges because "wedges are the solution to us, the clumsy ones".
Try again and post photos, please.