Since many operating systems provide filesystems with a snapshot capability I wonder whether it is possible to use this filesystem feature to ease the process of backup up the Jenkins data.

The table at https://en.wikipedia.org/wiki/Comparison_of_file_systems#Features gives a hint whether a filesystem supports snapshots.

No blocking or waiting for Jenkins being idle might be necessary anymore?

Problem might be to detect whether the underlying filesystem supports snapshotting and how to use it (without handling each filesystem separately).

This link might also be helpful.