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the tricky part about chains is that if they are used for too long in a worn / lengthened state they start to change the pitch of the chain rings and cassette sprockets
the longer distance between the pins will now sit in much shallower teeth that are now further apart
a new chain with short links will not be able to grip the shallow tooth shape
really badly worn chains do not sit fully on the chain ring only about 1/4 of chain ring will be engaged fully when compared to a un worn system
the chain will sit very loosely and can be lifted off the ring
wear to the three smallest cassette sprokets is common / too much torque will turn them into a deep shark toothed shape
the middle sizes of the cassette if worn will have shallow symmetrical teeth from many miles of gentle use
the largest sprocket will sometimes be worn into a an asymmetric wave shape, but I have only seen this on heavy duty MTB
on most bikes its a chain costs 1/3 or less of the price of replacing the rings and sprockets
so replacing your chain every 3000 km is more or less the same cost as replacing everything at 12000 km
and by then other parts will be worn I dont think may people keep their bikes after that point
as the cost of repair starts to approach the cost of new bike
