While you might think of the pool with all of its bonded metal as a sort of 'concrete encased electrode' for the building, connected with an 'undersized' EGC, the rest of your question doesn't make much sense.
You seem to suggest that in the event of a short circuit, electricity will be trying to find a path to the ground electrode. This is a _false_ understanding of how electricity operates.
Electric current always follows a closed path. This path goes from the source of supply, through conductors and the load, and back to the _source_. _Ground_ has nothing to do with this closed path.
In the event of a short circuit, a conductive path is established which bypasses most of the load, and excessive current flows. Again, this has nothing to do with the ground. The excessive current will flow in the service conductors feeding the home.
Where 'ground' enters the picture is that we intentionally connect the electrical system neutral to the earth via grounding electrodes. Because of this, the earth _may_ become a path back to the source. If you are standing on wet soil and touch an energized conductor, you get a shock because the electric current flows from the source, through the conductor, through you, then through the soil, to the grounding electrode, and back to the source. Not to the Earth, but back to the _source_.
Because we have grounding electrodes at the home and at the transformer, the earth will form a parallel path for the neutral. However the resistance of this parallel path is many orders of magnitude greater than the resistance of the neutral conductor. In the event of a short circuit in the home, most of the current will flow via the neutral conductor, with only slight current flow through the earth.
Significant ground current can enter the picture if the ground itself is part of the source of electric current. This is not the case with the electrical power supplied to the home, but can be the case when lightning current gets added to the mix.
In the event of a fault that causes significant current to flow into the earth via the various home grounding electrodes, it is possible that the pool will be the best grounding electrode, and that this could overload the EGC between the pool and the home. This could be caused by a primary to secondary fault at the transformer feeding the home, or could be caused by a lightning strike on the home electrical system. In either of these cases, the damage to the #12 EGC is the least of ones worries.
-Jon