"Can anyone recommend a Linux (preferrably cross-platform) tool like this which is appropriate for use in personal laptops?"

I think you should look into the 'borg' backup tool - it has become the de facto standard for remote backups because it does everything that rsync does (efficient, changes only backups) but also produces strongly encrypted remote backup sets that only you have a key to ... the remote has no access to the data.

The borg website is here:

https://borgbackup.readthedocs.io/en/stable/

and a good description of how it works and why you should use it is here:

https://www.stavros.io/posts/holy-grail-backups/