Hi JD, hope you don't mind me chiming in.
Sorry about those hogs lost :-(
For a proper safety ground, you need a separate bare run in the bundle from the main panel to all other sub-panels passing through each. The grounding rod can stay attached to the bare ground at the sub-panels, good for lightning strikes. And yes the bare ground sizing needs to be big enough to carry a short all the way back to the main neutral bar.
The main thing to remember is "never let the bare ground carry any load current whatsoever." Safety ground is meant to provide a virgin path back to neutral at the mains in order to quickly trip the breaker for any short circuit to equipment ground.
The problem with tying the ground to the neutral at sub-panels is that the ground now becomes part of the neutral carrying load current and eliminates the pure safety ground.