Currently we have one nice debugging() that should be used to receive and log as many information as possible.

But there are a bunch of uses of the direct error_log() php function instead. One quick grep reveals at least these (leaving apart 3rd part libraries):

admin/handlevirus.php
admin/report/courseoverview/index.php
auth/cas/auth.php
auth/cas/CAS/CAS.php
auth/cas/cas_ldap_sync_users.php
auth/db/auth.php
auth/db/auth_db_sync_users.php
auth/ldap/auth.php
auth/ldap/auth_ldap_sync_users.php
auth/mnet/auth.php
backup/backup_execute.html
backup/restore_execute.html
course/rest.php
enrol/database/enrol.php
enrol/database/enrol_database_sync.php
enrol/ldap/enrol.php
enrol/ldap/enrol_ldap_sync.php
lang/en_utf8/admin.php
lib/dmllib.php
lib/moodlelib.php
lib/simpletestlib/errors.php
lib/statslib.php
lib/uploadlib.php
lib/weblib.php
mod/wiki/ewiki/ewiki.php

So, I would propose to:

1) Modify the current $CFG->debugdisplay behavior to allow, "on page", "on error log" and "both"
2) Modify as many current direct call to error_log() to use the debugging() one.

That's all. Ciao :-)