Well, that shell script would be needed for Jenkins as well, except for the first three lines. In addition, it would not do the build if there are no changes, so that the dating of the files reflect when things last changed. And send e-mails to the guilty person(s) if the build fails due to compilation errors or whatnot. If all the developers were given access to its web interface, they would also be able to examine the logs and run builds manually to examine why the builds failed for platforms other than their own, or (less importantly) just to trigger release builds at different times.
So while Jenkins is a rather big tool with several features we don't need (and even more in plugins), I just thought such features out of the box would seem useful.
(Jenkins' new pipeline script system is indeed a lot to get familiar with, but I think they still support the old semi-point-and-click way of doing things.)