Is it safe to keep reusing a portion of old dough?
This is how bread was made in my area just two generations before, not only flatbreads but every kind of dough. And the process wouldn't be started from scratch with new instant yeast, because it didn't exist yet.
I understand why you would say that a diluted piece of the old dough would remain after a month, but mind that fermentation is a metabolic process that converts molecules, and although actual atoms could be there, they were processed and rebuilt each time the dough ferments.
Compare that with any plant you eat; a part of this plant was a previous plant that was used as a fertilizer, maybe even passed through the animal metabolism. It's a cycle.