This won't help not only due to what I said above, but also because $| refers to the currently selected filehandle, so STDOUT by default; it won't do anything for code like OP showed. Even though it won't help here, for completeness, $filehandle->autoflush(1) is one way of controlling it.

Yes. I'm always saddened to see unfortunate old Perl globals, such as $|, still being used today.

For more detail on this topic see: Re: what is the meaning of $| in perl? (Buffering/autoflush/Unicode/UTF-8 References)