Ok, here’s the question: should I run a #4AWG cable from the shack to the grounding plate to try to make sure all equipment is as close to the same ground potential as possible?
Are you looking for a lightning ground? A safety ground? An RF ground?
Wire gauge can play into the lightning ground, but it won't make a bit of difference for the other two assuming #12 or so.
A 30' ground is going to act like an antenna on every band, but what it does to you will change for different bands. If your ground plate is a quarter wavelength (or an odd multiple) from your rig, it's not an RF ground at all, regardless of the wire diameter.
Let's take 30 feet as an example:
30' is a quarter wave at 8.2 mHz and 3/4 wave at 24.6 mHz.
It could be very problematic on 12 meters and would probably not act like much of an RF ground on 40 or 30 meters.
I guess the next question is: Why do you think you need a ground?