For me, the simplest explanation would be that the pieces were essentially some kind of stones from the beginning as these would be present in some form almost everywhere. Whether pebbles or ‘skipping stone’ shaped rocks one might find in a river, such materials could be used in their found state. As you suggest, the grid might have in many cases simply been implied. This would be much more realistic for a smaller board size of, for example, 7x7 or less. Tic Tac Toe really doesn’t need much more than a bounding box to indicate the position, but it is just as easy to draw the grid as it is to draw a box around everything, but for a slightly larger board, a box might well have been sufficient.
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The practice of prisoner exchange also recognizes a very common reality which frequently occurs when using club or other publicly owned go sets : Running out of stones. They get lost or broken easily, are not looked after carefully the same way as personal equipment, and such sets rarely have 181 black and 180 white. Prisoner exchange is the obvious solution to continue the game. If not possible due to one side not having enough prisoners, it is likely a sign the game is over anyway.
More contemporary scoring systems also reflect this reality. You don’t really need that many stones unless you are filling in every point on the board, or have massive captures and recaptures. In fact, prisoner exchange is not even necessary most of the time, even if each side only has 125 stones. How many games have you played in person where the outcome was unclear after move 250? It is really only a very evenly matched game that puts demands on the number of stones in the end.
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