First, if the drive is optimized for quick removal, as in USB flash drives, no Recycle Bin is shown. Check Better performance, and probably enable write caching, for an external HDD or SSD.
Second, very large files or files that added to existing trash would exceed the size set for the Recycle Bin cannot be recycled. Set the maximum size of the bin appropriately. Even so, Windows may warn you a file is too large for the bin.
Third, as you mentioned, "Don't move files to the Recycle Bin..." should not be selected for the external USB drive. This setting is per drive.