Not quite. I had the quill that produces scrolls, and the chest that produces stones, and the meat cleaver that produces money, almost from the beginning of the game, and upgraded them a bit later on. I sandbagged a bit to make sure I got a lot of value out of them. I got lucky and got several scriptoriums over the course of the run. (And I got extra benefit from those by having “when you sell” items like the cat-chest that gains value when you sell, which is how I managed to get that much money.)
As soon as I got a cooldown stone, I used it on whatever highest-value cooldown scroll I had. Then, a few rounds from the end of the game, I used all the cooldown scrolls I had in order from highest-value to lowest-value.
Having played a number of games specifically aiming for minimum cooldown, I don’t think it’s too overpowered (as long as pain doll is fixed).
You have to consistently have a lot of money to search for items, and save all your cooldown improvements (potentially losing battles that you would have won if you went ahead and used the cooldown you had so far), and choose getting more cooldown over other things that you might otherwise get benefit out of. I had to dedicate essentially the whole game towards this goal in order to reach it a few rounds before the end, and sell or pass up other more powerful win conditions to get it (e.g. items to trigger axe charge, or rush items), and at that point it became the “win more” victory lap.
I think the bigger thing that might need looking at is the degree to which the items you find in the first three steps, before the first battle, vary so strongly in impact on the rest of the game that I tend to just restart the run if I don’t find interesting things.