Current state - tried again, after one captcha (two screens) on the first file it went to download nine other files as well, then asked for captchas again and can't go out of them.
Edit: looks like it's necessary to use a direct HTTPS view for a file, complete one or two captchas, download the file, several next files will work without captchas, rinse and repeat. At the end, find these "captcha files" and fix locations/names. If only I had an idea, why do download managers trigger such a weird reaction?
Edit 2: there is also an error "ERROR for site owner: Invalid site key", if all of this was not confusing enough already.