I think it is fine to acknowledge that in the big picture, this particular problem you're dealing with right now is fairly minor.
In some cases it is also ok to call someone else out, I think, e.g. if a group of friends came together to support one of the group who is having a crisis, and then another of the group monopolises their time by moaning about something insignificant. But this calling out is a bit tricky as it can verge onto dismissing someone's (real) problems and heaping guilt.
I think what OP is concerned about is not the 'putting into the big picture' aspect nor the 'belittling of someone's problems' aspect, but rather the language used, the particular phrase used. And I totally agree with OP that referring to the 'first world'/'third world' in this way to signifiy superiority, really, is out of date and bigoted.
There is a lot of language we use without meaning anything by it, but that does not make it ok.
It doesn't mean we 'can't say anything at all' for fear of offending someone. It just means that when we become aware of the problems with a particular expression we've always used without thought, we take it on board and find a different way of expressing what we mean to express.