Почему rsyslog перестает работать после rotate?
Добавьте в конфиг:
sharedscripts
postrotate
     /usr/bin/systemctl restart rsyslog 2>/dev/null || true
endscript

в логротэйт.конф добавь copytruncate

 Truncate  the original log file to zero size in place after creating a copy, instead of moving the old log file and optionally creating a new one. It can be used when some program cannot be told to close its logfile and thus might continue writing (appending) to the previous log  file  forever. Note  that there is a very small time slice between copying the file and truncating it, so some logging data might be lost. When this option is used, the create option will have no effect, as the old log file stays in place.