You said that hedging would be expensive for indices, I questioned this and asked for an example, but you neglected to give an example.
Since puts and calls are equally fairly (or let's assume unfairly) priced, then it doesn't make any difference for hedging (or whatsoever),
because when closing this options position then any inequality turns into a net equality, due to, as said, the Put/Call parity.
You're mistaking me with Amalgam.