I think the Snowhead theory is actually pretty solid but I would posit the irrational hypothesis that back when, a river flowed from Ikana canyon out past the graveyard, toward clock town and out to the Great Bay. This carved the canyon up which you travel after hookshotting up to the snag.
Eventually, a need arose to divert the river toward what became the swamp. The Ikanan Kingdom commissioned a tunnel to be built to shift the flow of the river to the south. Once completed, the tunnel dumped the river into a patch of land that hadn't previously had such a volume of water running through it. This caused the river to disperse, creating a wetland or swamp.
As you've surely noticed, the river dumps out into the Southern Swamp in far less volume than it entered the tunnel (as evidenced by the tiny ring of water near the Lost Woods at the river's terminus). As time went by, sediment lay down layers of strata over the silt making up the ancient riverbed.
Now, water follows the path of least resistance, which is what the river originally did: flowing downhill into the ocean across Termina Field. The river forecably diverted and channeled, the main flow emptied into the swamp but over manifold centuries since Ikana was in its prime, a branch of water reached out from the river's bulk and once again found its way downhill, following the old riverbed.
As the new branch trickled ever so slowly through the soft silt, it began compacting the soil around it, widening the cavity at a painfully slow rate. The more water that built up, the more pressure pushing the silt "downstream", the faster the cavity expanded.
The river eventually reached the ocean, either erupting under water from its subterrainian journey or filtering out through the sand once the surface beach dropped low enough to meet the water table.
Notice that the water beneath Clock Town in the areas below the Astral Observatory, Clock Tower, and the Laundry Pool seem to flow in that order: East to West, downhill, Ikana to Great Bay.
Obviously, this isn't what happened but wild speculation is fun!