People that accomplish this goal of showing no emotion ever resemble mannequins. So the ideal that you're talking about, really, is the "plastic ideal". This ideal was championed and proliferated heavily in the '80s.

The result from pursuing it exacts a very dehumanizing toll: when people cannot read your emotions, those closest to you misinterpret what you're feeling on a consistent basis. Miscommunications flourish. It is like being deaf, only it's your emotions that do not respond.

Being sensitive, on the other hand, may embarrass you a little bit in public (or around shitty people). But when it comes to your most valuable relationships, among those whom you truly trust, being sensitive is a superpower that rewards you over and over.