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I personally think it's a good thing for Cale that people don't know he has a healing power and how it works. Because then... People would get more demanding.
There are already too many people (background extras, not his allies) who look up to him like a saviour and hero and are grateful for being able to stand under the protection of his shield. The blood coughing and fainting is what makes them (the humane ones) not think of him as an invincible person who Should be at the forefront protecting them.
Imagine if people knew that when Cale Henituse coughs up blood, he's not actually hurt or dying but actually healing? That he's not injured and rather perfectly fine and healthy? There would be more people who would want to push him to become their hero and savior, relying on him to save them from all sorts of dangers. That is not something Cale, who has spent nearly half of his life sacrificing himself to save others' lives, deserves.
His allies too. Although I feel bad for them sometimes for all the headache and heartache he gives them, it's for the best.
People already view him as a hero and savior, a legend. They don't need more grounds to forget he's human. They'll take his protection for granted and push him to do more. Although Cale isn't the type to just do as they ask of him. In fact, the ones to make these demands would be the first to die. But it still would leave a bad taste in everyone's mouth, especially those close to Cale. Though they'd do everything in their power to destroy whoever says such things, it'd still hurt them to know people think of their Cale as more of a commodity than a person. Heck I'm afraid even some of the allies might get the idea that it's fine for the young master to cough up a little blood if it's for the greater good.
That's why, it's for the better that nobody knows the truth behind the coughing blood.