The “pursuit” of happiness is neither “required” nor an unalienable “right” (endowed by a “Creator”).
Happiness need not be pursued to be experienced. Happiness is an emotion. Humans can be he happy or not at any time both for and without any reason – regardless of what they do or do not do or pursue. Pursuit also does not necessarily lead to attainment. Anyone’s happiness or lack of it has more to with their own attitudes and choices than their “rights”. And with the possible exception of some artificially inseminated or cloned creatures cultivated from conception to “completion” in a laboratory, no human has been “created” since the Biblical “Adam” (and “Eve”); humans are “born” – and sometimes “again”.
Whether (or not) we are happy (once or again and again), on the other hand, is a “condition” humans can “create” – regardless of circumstances. Happiness is in many ways a habit – related to perspective. How we view things and the meaning we give them influences how we feel. The more we see and feel a certain way, the easier it is to do so. Regardless of personality, temperament, or circumstance, anyone can be happy – and the more often someone is happy, the more likely, easily, often, and longer he or she will be happy again.
Life and liberty are also not really (God-given) “rights” (or “properties”). Death is inevitable and can occur at any time. Ideas, ideals, and ideologies regarding freedom, liberty, and justice (for anyone, let alone for all) seldom seem seen in society. Tyranny, terrorism, slavery, and servitude are more the rule than the exception.
A “right” is what is “left” – defined and defended (as more than just a “privilege”); it must be protected to be preserved. A sense of “entitlement” (and “expectation” without action) often leads to the “erosion” of “rights” and the expansion of “wrongs” (either “experienced” or perceived to “exist”). While nobody may be “entitled” to “happiness” and the “reasons” anyone ever is or is not “happy” are entirely their own, no “right” or permission is ever needed (from others) to feel however someone may (desire to). With life comes the “liberty” to perceive and experience what we can – including happiness. Emotions and (imaginary) “rights” are more likely to be “elusive” and “at risk” when pursued rather than when practiced. Pursuit is optional – as is happiness itself.