Seconding that "Cogitations" is a great name for a puzzle game with these cog mechanics.
I'd love to read about your puzzle design process when you do get to it!
Playing with the prototype reminded me of Zachtronics games, and I think a game in that vein, with these mechanics designed around building machines that accomplish specific real-world(-ish) tasks given a specific set of pieces, would be very interesting. I think the limitations that Zachtronics games tend to set are also informative: everything is on a grid (usually a hex grid), and those pieces that can change size only do so in a small range and in increments that correspond to this grid. This limitation should allow the cogs' teeth to match up, since there would only be a small number of cog sizes and relative positions possible. Early in a Zachtronics game, you start with only a few pieces available with which to solve each puzzle, but as you progress, you earn new pieces required for solutions. You can go back to early puzzles and build potentially smaller (spatially), simpler (fewer pieces), or faster solutions using these new pieces. The games keep track of the complexity of your solutions along several axes and let you save your solutions for each puzzle, encouraging you to try different approaches. For the final few puzzles, you have every piece available and have to show a full understanding of their interactions to solve them.