It is important to teach children (and adults) healthy ways of interacting with people, so I am not suggesting to only throw out the idea of not hurting peoples' feelings, but rather, suggesting to replace it with a healthier, more robust way of thinking. The way I like to think about these things involves honesty and respect.
I think that it is important to teach people to be respectful to all people at all times, and to be honest with them, which includes being honest about your feelings as well as honestly asserting boundaries.
Most of the examples that adults want to use with children, that involve hurt feelings, can also be addressed by talking about respect and honesty. For example, stealing, bullying, or insults or malicious comments are all disrespectful.
I think honesty and respect make for a safer and more reliable moral compass than avoiding hurting peoples' feelings, because you can evaluate whether or not words or actions are respectful independently of how anyone reacted to them.