The problem is that if you keep running the worn chain, you'll ruin the large sprockets as well as the small ones. They are likely to be expensive. If the large ones are worn already, you might as well run the whole thing as it is until the sprockets go bald, since you'll have to replace them all anyway. The aim of the game is to keep from ruining the large sprockets, if possible.
Of course if the previous tenant had replaced the chains in a timely manner, the damage to the small sprockets would have been delayed a long time.
