The new QuietScrob version 500 from May 31, 2021 supports scrobbling of Apple Music tracks which are only streamed and not added to your Music library first. Check the "What's new" section on the App Store page:
• New look with easily accessible menu and settings, Dark Mode and Dynamic Type support
• Auto refresh, no more need to do it manually
• Scrobbling of non-Library tracks
• Loved tracks support
Also note that Apple Music will come up with lossless streaming at no additional costs this month. Creating Last.fm playlists e.g. with the templates for your personal radios and exporting them to Apple Music will enable listening to your Last.fm recommendations or other radios etc. in HiFi audio quality that way.