"I want Happiness" / "Remove `I` and `want`" story A man said to the Buddha, "I want Happiness."
Buddha said, first remove "I", that's ego,
then remove "want", that's desire.
See now you are left with only Happiness.
If the story given above true? If not why not?
If the story given above true? If not why not?

It's not canonical -- A man said to the Buddha, ‘I want Happiness.’ is listed as one of the "fake Buddha quotes". 

As for whether it's true, yes and no.

Yes to the extent that "wanting" is associated with unhappiness, though even there you might have to split hairs to distinguish between "unwholesome craving" versus a "wholesome desire for enlightenment" -- though even the wholesome desire is allayed once the goal is attained.

And no I guess it's not as simple as that -- it's like going to a doctor and saying, "I have cancer", and the doctor saying, "well take away the cancer then, problem solved" -- the Buddha was more helpful/detailed than that.

Also no because its "truth" depends on the definitions of various words, like "happiness", which are maybe just empty words.