láma anáchnu tzrichím chinúch?
I ask myself this question every morning when I hear the alarm...
seems to me we could understand "we" to mean people in general i. e. an impersonal sentence
I think you're right, in which case למה צריך חינוך? should be accepted.
Wouldn't that still be למה צריכים חינוך?
Either. Yours more so.
Actually, according to this, צריך is more common, just like AlmogL wrote.