As a former teacher, I do agree with the policy. School is about school and school events like ballgames, band competitions, etc. Private parties are not the school's responsibility and don't need to get dragged into school.
What the school really means by "hurt feelings" has to do with bullying, and believe me, it's alive and well even in our Christian schools. Try talking an elementary or middle school aged girl out of suicide and you won't be calling them wimps or sissies anymore. It may seem innocent to pass out birthday invitations, but in reality it is just another tool used by bullies to belittle and humiliate others. I've seen firsthand the damage done by Christian school bullies, and it's not pretty. As parents and teachers we have a duty to curb any inappropriate behavior used as a tool to hurt others. That doesn't mean every child having a birthday party is a bully--it just means we see a potential for division and hurt and want to avoid it.
As a Christian parent, I want my kids to learn Christlike behavior, which means treating others with kindness and respect. It includes not giving cause for offense when it can be easily avoided.