Personally I don't see the advantage of )}]]})} over ))))))).

It's more time consuming to close )}]]})}. With ))) we just repeatedly type ))) until the cursor jumps to the target opening paren we are trying to close. With a mixture of different parens we have to interrupt what we are doing and determine the correct parens which will continue the closing sequence.

I think there are people who value completeness- they enjoy working on systems with many rules tailored for specific use cases, and like accumulating knowledge about which rule is the correct one to apply in each situation; and there are people who value open-endedness- systems that have few simple rules that can be combined to obtain novel and unexpected results, even if these are not as optimized as they could be in a more complete system.

I picture the difference as that between a role playing game and chess: complex, domain-specific rules on one side, simple and abstract rules on the other.

Not really. You can't comment on this until you tried a lisp. I know that you haven't tried it because if you had you would have known about paredit and how it solves the paren problem.