I don't think your going to see the cable move. A new cable has no slop, a fraction of an inch at one ends moves the other end the same amount. If the same inner core cable was inside a tube with a 1/4 ID the cable would flop inside. Another thing, new cables have a coating on them which lubes and also takes up space. When the coating is worn off, there is more slop.
When the motor is running the drive shaft is spinning both upper and lower gears in the clutch assembly. When the rpms increase the gears spin faster and that rpm allows the clutch to move a tad more. That tad more allows it to move just enough to start the engagement and bang, its in gear.
Just replaced the lower shift on my Rinker, the coating was gone off the center cable. While mine wasn't bad like yours, mine had the issue if I reved it a bit to high in throttle only mode while warming up, it would bang into gear.
You shouldn't shift the Bravo with motor off because of the helical cut gears. This doesn't hurt the gears but the gears when together need rotation to come apart easy. The best way to get it in/out of gear (if time is available) is to jump the slave solenoid to turn the engine slightly as the handle is being moved toward neutral.
Is there any issue with going into reverse. If it goes in gear with very little movement, try rotating the barrel adjustment a couple turns forward