If this is all happening at a set time every day, couldn't it just be a cron job? You do have the script created already, CI is just running it for you.
Sure it could be a cron job, but using Jenkins adds some conveniences: a nice UI to browse projects, plus it archives the output of every historical build, locally-archived build artifacts, lots of extensibility with plugins, and so on.
Plus: Once you already got everything up and running you can take it a step further and deploy from the CI Server. In my previous job, the Ops Team loved it when I told them how to deploy from the CI instead of going to the "Release Developer" (Which was loathed by the Ops, because she was not the friendliest person :D)
It's worth adding that it does those things based on the output of previous jobs. Also, if you have many of these setups in parallel, it can create good dashboards and give you one nice (and free) GUI to take in the state of your build empire.
