How does earthing work when earth is dry?
There are no guarantees. Earthing systems will be worked out on the basis both of theory and of empirical results gained from long experience. The earth that you describe is extremely impressive, and far superior to what I have seen in some other standards.
Grounding does NOT ensure personal safety
Note that while personal safety is invovled in grounding considerations, the effectiveness of an earth is not liable to play a major part in improving many shock related outcomes and may make many of them worse rather than better.
The ability to handle fault currents without causing local ground potential rise and to thus trip power interruption equipment (fuses or breakers) is the major consideration. Within premises the path to earth for a person who contacts a live conductor will either be to a grounded metal object (kettle or toaster body etc, or via distributed local ground to earth - wet floor or apparently ungrounded semi conductive surface. In the case of a grounded appliance body, the grounding is intended to offer a short circuit to any fault current from within the appliance and will function without reference to the building ground, provided the return conductor is at ground resistance, or meant to be. eg in NZ (my country) we operate a MEN or "Multiple Earth Neutral" system where ground and neutral are connected at each switch board. Some systems may only connect neutral and ground at the building distribution box and in some systems there is NO neutral to ground connection - eg at least some shipboard systems float the whole system wrt local (seawater and hull) ground. In a ground connected system the local grounded appliance bodies will INCREASE the chance of electric shock for a person touching a live wire from another source than the appliance concerned as they offer a hard ground path, regardless of building ground efficacy.
In the case of distributed ground inside a premises, a situation similar to the above arises with current from an exposed conductor to ground being via the informal local ground and then to earth. Good building grounding may make the shock worse.
ie Building grounding will have little direct effect in protecting occupiers from shock. Where it does have effect is in ensuring that protective equipment operates.
ELCBs - lifesavers Where it DOES work is if ELCBs (Earth leak circuit breakers) are equipped. An ELCB detects the imbalance in current between phase and neutral (go and return) that occurs when a person diverts part of the current from the live circuit to ground. ELCBs are designed to trip at currents below that liable to be drawn by a person contacting mains. They are designed to trip in less than the time taken for one "heartbeat", thereby removing (theoretically) the ability to cause cardiac fibrillation. You can still feel the kick ! - ask me how I know :-). [[Back of clenched fist testing probably allows you to check this. YMMV. Don't try this at home. Ouch!]]
The above diagram is from "Electric Shock Protection"
Going to ground
Earth resistance is based on providing a means of accessing an effectively zero resistance earth that is "out there". "Out there" is accessed by providing a large enough connection to the zero ground that the resistance of the medium (soil) does not add too much to the resistance achieved. Often an "X" ohm ground is aimed at where "X" is set by experience as being adequate for the protection required. The described method of achieving "X" (here 3 x 20 foot rods etc) is based on acceptable worst case conditions (or should be).
A linear group of conductors spaced "not too far and not too close" relative to each other, form an effective cylinder of about the diameter of the bundle - with too far and too near being based on both theory and practice. This cylinder can be conceived to connect by "curvilinear squares" of the surrounding medium to a larger cylinder of surrounding medium which grows into an effective half sphere as you get further away. The resistance of each "square" is equal (when properly constructed) as a square which is N units wide will also be N units deep.
The transition from effectively a cylinder of conductor to a half sphere occurs over a few radii of the original conductor bundle. It's up to the specifying authorities to ensure that the typical water tables, soil types, conductor type, specified conductor arrangements and phases of the Moon are such that the arrangement will meet the need often enough to be safe enough for the applications considered. ie under very dry conditions with some soil types under some fault conditions results may not be good enough on some occasions. Cost and practicality play a part in determining how often "on some occasions" may be. As failure may lead to death or fires, earthing systems' requirements tend to err on the generous side of sensible.
When the earth is dry. sometimes, earthing works badly.
A pathological case for earthing is things on top of mountains.
The mountaintops tend to be pretty dry. People like installing observatories atop mountains. In a past life I worked with observatories.
The earthing rods are laid out in a ring around the observatory with a buried cable linking them. The rods have tops high in the air; point is to get some lightning protection, and an earth that works at all. (Lightning likes high, metallic things like observatories.)
The air is very dry on top of mountains. ESD is normally prevented by earthing.
Standard protocol was for staff to urinate on the earth spikes instead of using the porta-potty (whenever possible without upsetting tourists).
To make it harder, many sites share the mountaintop with radio transmitters. All that radiated RF is hard to screen when there is no functional earth to connect to. (Radio guys, you have the same problems but you caused mine dammit!)
A co-worker had similar problems earthing in a past life with sand dunes and earthing. Getting the septic tanks to drain on to the dunes can help.
tl:dr; Since the "earth" is a common factor to both you and the conductor, it's a non-issue.
It's not that the "earth" is dry and not very conductive at that point of contact, because if that were the case, why would my body be a better conductor, seeing that it's standing on some substrate directly above the "earth" that the copper/steel/etc is driven into. The main thing here that we're looking at is how much more 3 giant pieces of conductive metal want to take that current than your poor little body, and here, they want to a hell of a lot more.