Akeeba has been warning me for ages...
... about not using the default output directory for backups on all my J!3 sites. The same warning obviously comes up in J!4 as well, so I finally decided to do something about it.
It was actually really easy, as explained here on Akeeba's site. Add a folder, preferably outside your site's root, and point to it in the configuration of your backup profile.
I never even realized I could create folders outside of my site's root, but it turns out you can, no problem. Akeeba warns that on some servers, a folder outside the root could be unwritable, but my hoster's server took it all in it's stride.