Our Gemara on Amud Aleph describes the mitzvah of rejoicing on Shemini Azereth. The Peri Tzaddik (Chag Succos 43) explains the unique mitzvah of joy on this holiday over others:

The other holidays mention joy, but with some qualifying Language such as “Rejoice before Hashem” (VaYikra 23:40), or “Rejoice on your holiday” (Deuteronomy 16:14). However, for Shemini Azereth, it states “You will be only happy” (ibid 15), which Rashi interprets as a guarantee of happiness.

Peri Tzaddik explains that ordinarily it is not in a person’s ability to “produce joy in his heart” except by way of performing mitzvos such as Succah and Lulav. Yet, the special blessing of Shemini Azereth is that the joy comes with no dedicated mitzvah for the holiday. Rather, it is the accumulated effect of the entire Yamim Noraim season. The word Azereth, which means to hold back, can also mean a holding or accumulation of mitzvos.

The phrase Peri Tzaddik used to explain man and his relation to happiness struck me as psychologically significant. He said, אין ביד האדם להוליד השמחה בלבו רק ע"י מעשה המצוה a person has no power to generate joy in his heart without mitzvos. In actuality, people cannot make themselves happy by chasing happiness. Humans are hardwired to feel most happy and content when they are feeling the tension and desire for what they want, take on a fair challenge to obtain it, and then for a bit of time enjoying the satisfaction of achieving the desired goal. Yet soon it is a fact of human nature that without a new perceived need, and challenging effort to obtain it, dissatisfaction and restlessness will ensue. The secret of happiness is to constantly have goals that are fairly obtainable, that is the challenges within one’s zone of competency, and then to achieve it. Just as human beings need sleep and need love whether it makes any sense or not, human beings need to have the tension of desiring something and the challenge of working toward it in order to feel most happy when they finally obtain it for a period of time. If there are too few challenges, or the challenges are too easy or too hard, depression or restlessness is the result. There’s no escaping this fact of human nature.

One way to understand the nusach of the Borei Nefashos beracha, is that it makes this same point, see Tur OH 207 :

בורא נפשות רבות וחסרונם על כל מה שברא

who makes many beings and everything they are missing

We thank Hashem for creating enjoyments that we do not NEED to survive but still feel their absence and thus it creates joy to fulfill the desire.

Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation

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