kickdown cable making problem

First you need a light spring that pulls the lever on the trans fully forward at zero throttle opening.
Your lever doesn’t do that on it’s own.
Not having the spring on the lever holds your throttle pressure up making it seem like everything is fine, but it also makes the trans think you don’t need 3rd gear yet.

From the lever on the trans to the front cooler line works good.

Put the spring on it and I’ll bet it solves the problem.
Exactly. I had to put a spring on mine to fully retract the lever. The throttle return springs on the carb were not enough to make that happen.