Why is my garage grounded ONLY when it rains?
To me it sounds like the conduit was used as the grounding conductor, as allowed by code, and it rusted through, therefore it works when the ground is wet and you have a ground connection through the soil and water.  If this is the case, then an additional electrode at the garage probably won’t help as the pipe is in the ground.
Your electrician wanting to dig it up makes sense to me because when debris get in the pipe, it becomes impossible to pull new wire. With current code, the ground can be run separately, but that would be pushing the code in my opinion but legal. You can run a ground as long as they originate from the same panel.
Earth is a horrible conductor. I see high resistance quite often; this is why I don’t think another rod will help.