Depending on how much inaccuracy you can tolerate, you can get a reasonable facsimile of stripping all HTML by doing:

$page =~ s/<[^<>]*>//g; # Note the added < inside []
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assuming the entire page content is in $page. A line by line approach like that in your original post will fail on tags that span multiple lines. The regexp above will break if you have unbalanced < or > inside of html tags, but may be good enough for your use.