It's a stupid headline. Oxford's published statistics show that grammars are just as likely, overall, as independents, overall, to get pupils into Oxford. Some maintained schools have a dismal track record, which various initiatives are designed to ameliorate at the point of entry (UNIQ, the flagging system etc). Some independents (including some big names) do abysmally ( especially given the cost of the fees and the aspirations of the parents), while some always do breathtakingly well, with 50% of leavers gaining places at either Oxford or Cambridge (St. Paul's Girls', Westminster etc.). The crucial factor is about the level of selectivity of the individual school.
People talk too often and too disparagingly of the lack of parental support for children at state schools. It may be well meaning but it's insulting to parents like myself, with three children at or soon to start at Oxford, technically eligible for FSM. Having watched my daughters and others go through the system at Oxford in three of the last four cycles of interviews it strikes me that the system bends over backwards to be fair and to select solely on merit. The re-introduction of grammars generally, with a decent alternative this time for those not able to get into or not choosing the grammars, would do more than anything else to address the inequalities referred to above.