Tony has been in the public eye as a businessman for a helluva long time. Like Steve said, he was a very public figure and knew how to play the part of charming celebrity, but the actual PR part was never anything he dealt with, and Bucky never came near that realm of handling people. Although I was very happy that Pepper chimed in, because she's been in the business just as long, and Tony has kinda proven to be a nitwit about it time and again.
(I have plans for them to meet up eventually! I can't wait for that conversation. XD Also, when I was waking up about ten minutes ago, I was hearing Tony say he was consulting Bruce about some of those drugs. HIVE MIND GO GO GO)
I wouldn't be surprised. Psychiatrists were just starting to think of mental illnesses as having a physical cause in their day, so that concept was really new and not generally accepted by the public yet. (Let's be honest, we're still struggling with that.) And even if you did accept this, those were still the psychiatric dark ages; that was about the time that insulin shock and ECT came into being. And the field had been using bindings forever at that point. I don't know for certain if lobotomies had been invented by the time Steve and Bucky 'died', though I am fairly certain they had, because they went mostly out of style in 1953, when Thorazine was invented. So take all the stigma we have about mental illnesses today and multiply it by a factor of OVER NINE THOUSAND.
After helping Steve all their lives, Bucky's pride about physical ailments is fairly low, but mental health help was just not something you sought out. Not even for the seriously ill, like schizophrenics.