it is not the offender's job to avoid offending people, because that is completely unavoidable.

If the offender wishes to avoid the consequences that come from the offended individual successfully portraying them as offensive or otherwise doing reputation damage then yes, it is the offender's job not to offend people. It is likewise the offender's job to avoid offending people if they happen to intrinsically value other people not being offended by them, where the degree of value is less than the cost of limiting their freedom of expression and extra modelling of expected reactions.

It is instead the offendee's job to realize that being offended means that one has a bias/image, and that the only way to not be offended in the same way again is to remove that bias/image from one's own psyche.

Not true. Have the offender publicly executed via gruesome torture. That can make future instances highly unlikely. (Reduce the degree of 'tantrum throwing' as necessary.) You can change the environment to be more like how you want it to be.

It is also worth noting that avoiding the emotional experience of being offended is not the point. In the same way avoiding the emotional experience of being angry, sad or afraid is not the point. Those feelings are usually there to indicate that we are best served by taking actions in the actual real world so that outrageous, sad or scary things don't happen to us as much. Removing the feeling without removing the stimulus is not always a good thing.