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Loggers don't deal with output - handlers do - so I would prefer not to add an output-related method to a logger: people confuse the two enough as it is. Note that 
stream = LoggerWriter('foo')
gives an equivalent result to
stream = getLogger('foo').debug_printer()
and in my view the former is preferable.
I agree that there is no compelling reason to replace sys.stdout, though it means that you have to pass file=stream in each print() call.
It also makes more sense to use print() rather than have a separate API that looks just like print(), as you essentially would have to duplicate the functionality of the print() API and track enhancements to it over time (to avoid surprises).