This is actually the "right" way to do this, or at least the way sourdough bread has been made traditionally almost everywhere I've looked into.
The same kind of process applies to yogurt. You make the starting batch of bacteria in a similar way but then you just use a portion of the previous batch.
Yes, lots of people in India make yogurt (called curd here) at home daily in that way - just add a spoonful or so of the previous curd to the warm milk of the next batch to be made. I doubt anyone adds any separate starter culture - the old curd plays that role.
