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Most of the time when I'm working heavily with exceptions it's something related to contextlib, so I'm likely getting my exception details from either sys.exc_info() or the arguments to __exit__. That means I start out with an exception triple, and the only time I need to look at type(exc) or exc.__traceback__ is when I'm following exception chains.
However, Antoine's right that if you got your exception from an *except clause* rather than one of the more indirect APIs, then you're just going to have the exception, rather than an exception triple.
So I think that's where the argument for accepting "exception-or-triple" comes from: handling an exception directly is for use with except clauses, while handling triples is convenient for use with sys.exc_info(), __exit__ methods and for general backwards compatibility.