I think people are mostly able to be outraged at war and famine and genocide and at the same time be able to articulate dissatisfaction with issues closer to home that may not be life or death matters but are none the less important if we want to create a fair and decent society.
The one does not exclude the other.
If you are raising your children to only be concerned with big world problems and not to also think about issues of fairness and decency, then you are doing them a disservice, and underestimating their intelligence and capacity for critical thinking.
It is perfectly possible to think about large issues and smaller ones and take a stance on all of them without becoming overwhelmed. It generally comes naturally when you have worked out your own belief system and moral standards in life. It is not as if people sit around pondering what to think about certain things, most people instinctively know whether something is wrong to them because they have a firm inner integrity that they have built up throughout their formative years.
And those beliefs will include positions on sexism, racism etc. the fact that something may be a small example of sexism in a limited arena doesn't mean you can't acknowledge it as such and refuse to accept it without becoming overwhelmed.
And of course if people stand up against small examples, those who propogate this kind of insidious bigotry will have to pull thier socks up and stop using it. And as such it hopefully will occur less frequently and your children will be less in danger of being overwhelmed by thinking about it.