Eventually I tried out the sync mode. But I realized, all of my Queries and Templates on pages do not work (I think, these are called “Live Queries”) . Only the raw codeblock is displayed.
Are Live Queries intended NOT to work in sync-mode, anyway? Or do I have a problem here?
I tried it locally, with different browsers. And even “externally” with https via reverse-proxy (nginx).
Thanks a lot for all hints!
This should work perfectly and I use this effectively full time. You may need to wait for a full sync to finish, which can take time if you have a large space.
Are you seeing any errors in the JavaScript console?
Thanks for the hint, Zef!!
That pointed me in the right direction 
Turns out, there was indeed a javascript error listed.
Background:
During my evaluation of SB I also tried to install the graphview plugin. Later on I realized, that this PLUG is known not to work anymore, so I removed it from the PLUG List. However, it looks like either the js-file was not removed from the _PLUGs Folder or it was added again during the sync. Anyway - silverbullet found an error in the JS-file (which was visible in the JS-console) - and seemed to stop using the other JS-files, too.
After deleting the broken graphview.js, syncing and Reloading everything works as expected!
Only interesting to see, that everything was working in online-version, but not in the offline/sync mode… (I suspect, the broken js file was still in the offline DB, but not online…