Detached Audio comes back into the Exported video in iMovie

Very disconcerting, for sure.

So, I tried your procedure of detaching and deleting the audio and then exporting the project. It exported fine with no audio. What exact steps did you use to detach and delete the audio? Can you send a screen shot of your project screen that shows what you see after doing the detach/delete procedure? Is there perhaps other audio in there that you did not delete?

Some tips:

1. If the audio contains sensitive information, and is not otherwise important to the video portion, then best to record without a mic or with the audio muted. If necessary, you can always add an edited audio clip later.
2. Although it is the safest procedure other than recording silently, you don't need to detach and delete the audio before exporting. You can simply mute the audio and then export. (Not sure whether the recipient could recover the audio with special software.)
3. And, of course, before sending to a third party always first play an exported video completely through to make sure it contains no objectionable material. Assuming that you retained copies of the vids that you sent, you can play them back on your computer to confirm that no objectionable audio was included.

-- Rich

It looks like you did everything correctly to detach and delete the audio, and export the project. I can't explain why the deleted audio reappears in the exported movie. That being the case, iMovie is not working correctly.

Try deleting preferences. To delete preferences, open iMovie while holding down the Option and Command keys and select to delete preferences in the box that appears. iMovie will open in a new library. Reopen your old library to get back to your projects.  Then export some test videos to see if the deleted audio remains deleted in the final product. Deleting preferences is a safe procedure that will not cause data loss or disruption to your project.

You could also try booting up in Safe Mode (see Finder menu for instructions) and, after everything loads, immediately shut down and reboot in normal mode. Then see if iMovie works properly.

If none of the above solves it, try the following diagnostic procedures.

Updating iMovie

To update/redownload iMovie 10, drag your iMovie 10 app from the Applications folder into the trash, but do not empty the trash. Sign in to the app store and open your purchases folder by clicking on your name at the bottom of the app store side bar.   You will see your purchases displayed, including iMovie 10. Redownload/update iMovie from there. If not successful, drag your present iMovie app out of the trash and return it to the Applications folder.

Resetting the System Management Controller

Try resetting the System Management Controller (but no longer available for the new M1 computers):

How to reset the SMC of your Mac - Apple Support

-- Rich