Your design closely follows the concept of the Montessori learning tower, which has been around for many years.
This ad includes enough dimensions to replicate the original. Since the original has been tested for many years and proven stable and safe, I would probably try to copy that proven design as closely as possible.
You will note that the assembly is by Confirmat screws. Confirmat screws require a special stepped drill. There are various sizes, but 5mm and 7mm are the more common ones.
Confirmat screws have two significant advantages.
1. They are designed to screw into the ends of sheets of plywood, MDF, particle board, etc.
2. They can be installed and removed multiple times without losing any holding power. So the adjustability will remain.
3. They have vastly more racking strength (side to side wobble strength) than almost any other mechanical fastener. An important consideration for this tower.
The RTA manufacturers association (ready to assemble—think “IKEA”) contracted a testing firm to test a variety of fastenters/joinery systems.
It was intersecting to note that dadoes add almost zero strength to the racking. The dadoes in this illustration apparently are for location purposes only. The Confirmats are supplying almost all the joint strength.
You could choose to use dowels or dominoes for location purposes along with the Confirmats.
I get mine from AH Turf—but you need to assemble a significant order to have that make sense. Amazon sells smaller quantities at a higher cost. The step drill has to match the size of the confirmats.
Do not try to convince yourself that construction screws or drywall screws are “just as good”. They are not.
I agree with other who mentioned that 1/2” thick seems too thin. Mostly it will be too thin for any worthwhile fastener. So stick with 3/4”.
I have seen ads for Montessori Towers made from solid wood and others from “highly compressed pine products” (probably particle board or MDF).
Lowes sells glued up 3/4” thick pine in wide planks, if you want to go solid wood.
I would probably go with plywood. I would round over the edges and fill them with wood filler or grain filler for a smooth appearance.
https://busywisy.com/products/kids-step-stoolNote: I could not find any ads for the original Montessori towers. I had seen them when I went with a friend to the Montessori school for her daughter many years ago.
There is a “genuine” Montessori School learning tower. I don’t know how to find that information. The example I linked seems similar to what I saw all those years ago. If you can find an original example to copy from, that would be excellent. It is a classic.