I wouldn't say it's about denying happiness as much as it is about not allowing to lose reason.
wiki said:
One must therefore strive to be free of the passions, bearing in mind that the ancient meaning of 'passion' was "anguish" or "suffering", that is, "passively" reacting to external events—somewhat different from the modern use of the word. A distinction was made between pathos (plural pathe) which is normally translated as passion, propathos or instinctive reaction (e.g., turning pale and trembling when confronted by physical danger) and eupathos, which is the mark of the Stoic sage (sophos). The eupatheia are feelings that result from correct judgment in the same way as passions result from incorrect judgment.
So one of the components was the so-called equanimity, or psychological composure.
Complete stoicism with all of its outdated and idealistic assumptions appears unachievable, while the basic points can manifest in individuals that we can attribute to this philosophy.
Becoming stoical, or adopting it fully is comparable to adhering to any other set of beliefs and ideals, quite improbable and unrealistic to perform. While people can be rational or logical, people will manifest other, sometimes opposite qualities, despite their path being more convergent to this or that set of values when viewed as the entirety of lifetime.