I think it comes down to a utopian vs non-utopian view. The right have strong values, but they position them differently.
People on the right tend to see social problems, including justice issues, as being complex, often with no perfect solution, and involving pros and cons in terms of solutions.
It's like Thomas Sowell said, there are no solutions, only trade-offs. That might be a little strong, occasionally there might be a simple achievable solution, but the fact is that's rare. So for a conservative, the political process typically means figuring out what is possible in the scenario you have, and weighing the pros and cons, and coming up with a pragmatic approach that will work, or at least help, without causing too many problems of its own.
People on the left tend to think there is a possibility that there can be some kind of final just society, that if only we enact the right policies, you will have the outcomes you want, they often think it is social policy rather than human nature (or physical nature) that is the cause of the problems, and if you have the right "true"policy there won't be trade-offs.
(And I know that sounds crazy on the face of it, but when you look at a lot of the policy things many people on the left (not centrists typically) talk about, that is the approach. Like the OP - bombing kids is bad, but can't seem to address the facts of war where it's pretty much inevitable, even if people try not to do it.)
I think the other things that comes into it is that people on the left tend to recognize only a few moral axes, mainly fairness/unfairness. Whereas conservatives typically recognize a few more, and weigh them differently. So the people on the left don't think some of the moral considerations of conservatives are "real" they interpret them as some kind of selfishness.