I suggest you go read art 250 part III from start to end, it is not terribly wrong (edit: I meant long here), but there is a lot in there, maybe seeing something will trigger something in your mind and either give you a better understanding or make you ask the right question for something you don't understand.
Otherwise the two main functions of the GES is to provide a earth reference for the "grounded conductor" and to provide a place to bleed transient voltages (especially from lightning) to. The importance of the GES IMO sometimes gets too much emphasis, but it is not exactly easy to determine just how important it is or isn't either. I once had an inspector put me down like I did something terribly bad because I hadn't yet connected the GES on the project, and told me how that can have terrible results should there be a lightning strike - I let him rant on but in my mind there wasn't going to be much $$$ damages should this happen, and if it were a direct lightning strike to the facility, there was going to be damage with or without a GES.
As far as electrodes go - 250.50 starts off part III with "All grounding electrodes as described in 250.52(A)(1) through (A)(7) that are present at each building or structure served shall be bonded together to form the grounding electrode system." Note it says "electrodes that are present" you suggested that you understand it as "all grounding electrode conductors present" - not the same thing.
The first three items are items that may be present without the electrical designer/installer mandating anything to be present. If they are there you need to use them. The next four items are primarily designer/installer options, they will not be there unless part of design or specification. If none of the seven types are present you must make at least one be present and that is usually going to be one of the last four, with the driven rod being most common and most economical.
Now move on to 250.53 and it says rod pipe and plate electrodes need to be supplemented with another electrode unless they are 25 ohms or less - this is because it is typical for rod, pipe and plate electrodes to have a higher resistance then a water pipe building steel or a CEE. When a rod is the only electrode, many will just drive the second rod automatically because they don't have the means to determine if the first rod is 25 ohms or less and driving the second rod is less costly then testing the first rod anyway.
The water pipe electrode needs supplemented not because it is a poor electrode, but because metal water pipes are often replaced with non metallic pipes - and if that were the only electrode before the change - you are left with no electrode at all.
So in general it boils down to if there is a water pipe, (qualifying) building steel, or a CEE present you must all of them that are present. No supplemental electrodes required other then a supplement if water pipe is the only electrode present. A CEE or building steel does supplement a water pipe if it is also present. All other electrodes are generally optional, but at least one of those option must be included if there is no water pipe, building steel or CEE.
