The real problem is that the server is working with /cgi-bin as the default directory, instead of the public_html folder above it, as it would with any regular websurfing

Have you looked into the possibility of moving your script outside of cgi-bin and into public_html? In my experience, just about every shared-hosting Apache servers are configured to run CGI scripts anywhere, if they have the .cgi (sometimes .pl) extention. If it's not, have you tried adding AddHandler cgi-script .cgi to the server's configuration files or to a .htaccess in public_html?

While this may not be the answer you were expecting, it will save you lots of work.

Update: Rephrased a bit to clarify in response to sintadil's comment about default settings. He also mentioned the security consequecences of my suggestion.

In reply to Re: Scripts to recursively reading in HTML files by ikegami
in thread Scripts to recursively reading in HTML files by mwhiting

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