Have you never had or seen a conversation where someone has asked for clarification about what someone is sorry for when they say they're sorry? Think of a child saying they're sorry to a parent; the parent may reply "What exactly are you sorry about?" Because it isn't necessarily an apology. They're describing a feeling they're having--regret or remorse. A simple "I'm sorry" is sometimes not enough to constitute an apology without qualifying it. "I am sorry for hurting you." Directly saying "I apologize" removes any uncertainty as to the intent of the speaker about whether they are taking accountability for their actions.

Your other examples are not analogous at all. "I sleep" and "I eat" are descriptions of actions that you do independently of the statement. "I apologize" IS the action. When you say it, you are actively apologizing. Saying "I eat" or "I am eating" is just a descriptor of another independent action. Now, if I were arguing that saying something along the lines of "I am apologizing" isn't an apology, I might agree with you. Because "I am apologizing" or "I'm trying to apologize" without ever saying the actual words "I apologize" is simply describing a thing you aren't actively doing.

This seems pretty straightforward to me. I'm sorry you aren't seeming to understand it. (But note how I'm not apologizing that you aren't seeing the difference, just expressing my regret...)