I agree with you here. I think a reasonable example is the Wayback Machine. Very useful for archiving web content that has disappeared for whatever reason (usually lapse of web hosting). But if site/content creators want their content excluded, the Wayback Machine operators are very responsive and will stop hosting this content. I anticipate that asking for your content to be excluded from training sets after the fact will be much less pleasantly received, as the model would have to be relearned and this is expensive.