Schoolchildren would have to stop using any service that doesn't have a block-anonymous setting, or, accept that the school will ignore any complaints about anonymous messages. I think the biggest problem with the whole thing is that the kids won't stop using the services they want to use.
Apple could theoretically make a system setting to block unidentifiable messages (not phone calls) specifically to protect kids, in all apps on that phone.
When you speak of erasing the concern, do you mean the concern about bullying or the concern about going into children's phones and looking at text messages?