No. Backups from the GUI are either scheduled or require starting with Run now button.
One way to have scheduled backups sort of do this is to only backup if drive is present.
https://github.com/duplicati/duplicati/blob/master/Duplicati/Library/Modules/Builtin/run-script-example.bat
(also available in your Duplicati installation) shows how the exit code can cause a “don’t run” of a backup.
Scripting options such as run-script-before would be used. As usual, don’t remove drive during a backup.
offers another solution which might be closer to your instant start. To integrate with the GUI (which can be simpler to use for operations such as restore) you can Export As Command-line, and run from that event.
Device mount detection (USB or otherwise) has other ideas as well, and I can’t say which is best for you.
If you want multi-drive rotations, your life gets more complicated. A backup job is paired with a destination, however there are some ways to handle it (and several forum posts) if that becomes necessary for you…