The problem appears to be with exporting as "8 bit RGB with ROI" - presumably to keep the "ROIs" on the image the annotations have to be switched back on (which includes the "not for medical usage" text).
Therefore Osirix can still be used to anonymise ultrasound by slightly modifying the instructions…
1. Use the rectangle to select the image and exclude the patient data on the 1st image of the US series (the not for medical usage text comes back up, ignore it for now)
2. Click the shutter button
3. Now click 2D viewer -> annotations -> none
4. Click file -> export -> export to DICOM file(s)
5. Select all images of the series
6. Select "as stored in memory in 16-bit BW" (rather than "8 bit RGB with ROI")
The new series will have the anonymised ultrasound images (in colour, despite having selected 16-bit BW) as a separate series.
When manipulating a single image (e.g. straightening and cropping a plain X-ray) as long as you select "as displaced in 16-bit BW" or "as stored in memory in 16-bit BW" it also works as required without the "not for medical usage" logo.