There are many charities working provide more schools and train teachers and I don't believe Xprize, or anyone else is suggesting we should eradicate teachers.
The Xprize competition is about using technology to supplement existing teachers where they exist and act as a substitute for them when they are not available until we can train more to do the job.
You are correct in saying dealing with the problems of poverty is a multi-layered subject. We need to provide access to clean water and sanitation, education and the ability to trade fairly so they stop being reliant on charity. This is just one component in this joined up approach.
250 million children have no access to education partly due to need for many to walk several hours a day just to get water and partly due to a lack of suitable schools and teachers. Charity: Water is working to fix the first problem and we at Bad Voltage have so far raised $770 to help them. XPrize have realized that to meet UNESCO's figure of 1.6million more teachers needed by 2015 is going to be tough and are asking can we use technology to help here.
I don't know if it can and I fully accept there is no substitute for a good teacher. But, this has got to worth trying, hasn't it? Also note, this is probably not going to be an online solution since we can't assume the children involved will have any internet access though I'm sure the proposed solutions will make use of the internet if it is available.
@jonobacon please correct me if I've misinterpreted the aims of this Xprize in anyway.