I doubt the transmission is related to this. The cruise control is just a cable and servo that holds the throttle plate and the needed position to hold the speed.
If the switch turns the light on and it comes on but the speed does not set then the issue is most likely the servo which is what moves the cable to the proper position.
I attached the complete process below on how to replace this as this is what I would do and retest the system. Let me know if you have questions. Thanks
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Thursday, March 4th, 2021 AT 2:30 PM