As a manager, and I have no compunction telling my team to try and fight their own battles first, and if they need to refer someone to me to resolve the issue,

As a manager, and I have no compunction telling my team to try and fight their own battles first, and if they need to refer someone to me to resolve the issue, then to do so. I'm there to support my team 100% at all times as well as to deliver business outcomes. If you're getting upset over issues like this, perhaps management is not for you. Of course it may also depend on how tenuous the relationship with your own manager is.