There was certainly quite a bit of deep talk about "integrity" and "character" in our grandfathers' generation, that was ultimately relating to issues we would now comprise under so-called 'mental health'. It's not clear to me that this medicalized framing ("...health") is necessarily and consistently better than a more traditional one focused on developing a well-adjusted character.
Integrity and character are about values and how you plan to behave and expect to have others behave towards you. They are not the same as your ability to process emotions that emerge as a result of that behaviour.
Having values is important. Integrity, humility, all of that, absolutely useful.
They are not in themselves sufficient to assure you of good mental health.
We care about the smooth processing of emotions, among other reasons, because when impeded it generally affects how we're going to plan and behave; especially when under some sort of stress. This is not something new to our generation; philosophers have had a clear undestanding of this for millennia, in both Western and Eastern philosophical traditions.
We care about how we plan and behave, because we feel emotions about things that happen. Like you say, nothing new.
Some other old-fashioned-y terms in this context: "strength" and "fortitude".
Men generally process negative emotions in private so others don't worry about them. This has led to the incorrect common viewpoint that men don't process these emotions at all, and attempts to make men process them like women do.