I actually do not agree with the idea that apologies cost nothing.
Where there is pain there would ideally be learning. But how is this to happen if reflexively-offered apologies invite those who hurt to always see the other as at fault?
Emotional strength is the ability to laugh at or rise above any amount of bias, prejudice and bigotry — seeing these things for the stupidity they are. But perpetually being given the idea that your hurt has external causes only saps at your ability to develop such emotional strength. Where is the motivation to develop yourself in this direction if the hurt is routinely portrayed as resting with other people?
There is an argument that reflexive apologies just leave individuals too emotionally vulnerable to cope with the nastier sides of this world. Are such apologies really a kindness given that, as a result, the individual can become even easier to exploit, intimidate and hurt?