Stringing the error clusters together to force an order works because LabVIEW is data driven. That means a vi or case statement or loop will hold up execution until all data inputs have arrived.
The error cluster as an input isn't necessary for any of the vi's, but the vi will wait for the data in the error cluster to arrive from the last vi before it will start execution. The error cluster output also isn't "released" until the vi producing it completes all internal tasks, so strung together like that one vi has to completely finish before the next one will begin.
If they stand alone they will execute in no specific order, essentially simultaneously.
You have what seem to be two independent operations:
I'd probably keep these two separate as you originally had them, but give each their own While loop or sequence structure. There's no particular reason to have the Arcade Drive inside the kicking While loop, since it just keeps repeating the same identical action over and over. Separated, the drive code can be stopped after a fixed amount of time has elapsed, rather than continuing forever. You can also add a time limit to the whole structure as it currently stands and stop everything at once.
It appears that your motors will start at full speed, but will never stop (until Teleop takes over presumably). You'd go from the rear zone to the front zone until you ran into something to kick if a ball was missed or the sensor broke. I imagine your partners would be upset if you launched off a bump and came down on them at full speed.
It's not shown from your attachment how the outputs of "auto kick and solenoid logic.vi" line up, so we know what's going to which case statement in Autonomous Independent.vi.
It looks like your timing delays don't delay, because for the places where you use feedback to pass times back through case statements for comparison to the current time, no initial time is ever set. I'd guess that the unset time is actually always zero, so the False cases will always be executed.
I'll take your word for it that all those "auto kick and solenoid logic.vi" feedback and feedforward nodes work the way you want them to. It appears you have a paradox where you are checking the output of a case statement before it ever gets executed. It'll always have an indeterminate value at the time you go through those AND checks.