The first Perl script in my original post had 'text/plain'; that's the one that I'm talking about.

And as blahblah correctly points out lower down, this appears to have something to do with CGI, because if I remove that dependency as follows:

#!/usr/bin/perl -w

$|=1;

{
    print "Content-type: text/plain\n\n";

    print "Autoflush is currently " . ($|?"on":"off") . ".\n";
    for ( 1..5 ) {

      print "Value is $_.\n";

      sleep 1;
    }
    print "Autoflush is currently " . ($|?"on":"off") . ".\n";
}
[download]

The script works properly. So CGI is ignoring $| and doing its own caching. Now I know.