Basically, does the success of ZEC rely on the success of BTC or would it be able to replace it?
imo, both... right now ZEC is somewhat tied to BTC, but that'll prolly slowly end overtime.
will it replace? possibly, long term. ZEC is basically cheaper BTC with the option to encrypt your TX, larger blocks, faster blocks, and ASIC hostile ...pretty-much depends on what the markets value!
Other networks are much less like a dysfunctional family and are doing what it takes to go mainstream and actually be competitive with old money and banking systems.