I wouldn’t characterize the Buddhist approach as “achieving happiness by reason alone”. It’s not a thought process. It comes from insight into the nature of one’s self and things.
Unhappiness (Buddhists use the Pali word “dukkha”, which roughly means “unsatisfactoriness”) is a result of overly identifying with the ego. The ego is a thought process, a psychological division between an imagined self and an external world. The self that comes of that division is necessarily a troubled one; it’s a tiny, fragile speck in a vast universe of flux. Existential angst is part of the package. The self lives its brief life struggling to consolidate its position (a hopeless task), chasing after sensory pleasures and distractions, and trying to keep pain at bay. It becomes an endless game of “I’d be happier if only This or That were different about my life”, but that never actually turns out to be true in the long run.
The core of the problem then, from a Buddhist perspective, is that one is invested so completely in the welfare of that tiny self, which is little more than an idea. The idea of self has some provisional utility, but mistaking it for who we really are and living in service to it is a gross misunderstanding.
Happiness is release from that misunderstanding. One can’t reason one’s way to that release. There are plenty of modern thinkers who acknowledge the fundamental unreality of the self. but nevertheless live to serve it. The illusion is too deeply rooted and habitual to be reasoned away.
But leaving aside the methodology of release and to more directly answer your question of how that release affects behavior in the world, breaking through the illusion of self radically alters one’s perceived relation to the world. Acquisitiveness—the continual struggle to find what’s missing in one’s life—gives way to generosity. Self-serving gives way to concern for the welfare of others. What’s important is no longer satisfying one’s own desires, so all of the energy that formerly went to that endeavor can be redirected to the benefit of everyone.