It isn't clear to me that the connotations of "oversensitive" as it's used here are justified. Some people suffer, to greater or lesser degrees, in situations that I don't. That doesn't necessarily make them oversensitive, or me insensitive.

There are some things we, as a culture, are more sensitive to now than our predecessors were. That's not necessarily a bad thing.

I don't believe a rational person, in a situation where honesty causes suffering, necessarily prefers to be honest.

All of that said, I certainly support discouraging people from suffering, given the option. And I support discouraging people from claiming to suffer when they don't. But I don't support encouraging people to keep their mouths shut when they suffer. And I suspect that many social structures that ostensibly do the former in reality do the latter.