Why doesn't ::Stream set it after it has finished printing stuff to STDOUT automagically?

Its none of its business ?:)

I really don't know, this Bug #37038 for CGI-Application-Plugin-Stream: PATCH: improved auto-buffering and error handling. indicated it should have been done in 2008, go figure

Shall I do ... in my runmode?

You may :) You may also write that as

{
  local $|=1;
  ...
  return;
}
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Or does it mean I should switch the buffer mode in general for the whole app by stating $| = 1; in the setup() stage of CGI::App??

It may. You could even

{
    local $|=1;
    my $app = MyCGIAppSubclass->new;
    $app->run;
}
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