Michael Shammas' call for philosophy as a requirement in lower education is his argument on why he believes philosophy is necessary for a healthy society and a strong republic; "because the capacity to debate requires the capacity to think." He extols philosophy for encouraging students to "entertain a thought without accepting it" and be open to the "possibility that one is wrong."
He wonders why philosophy is not already standard curriculum, supposing that "perhaps the subject seems too esoteric or pretentious" or that "[it] could encroach on the sort of questions religion purports to answer." I think Shammas is missing two key elements of educational history. The first of these is the Educational Progressive movement, which by the 20th century had already branded rote memorization and repetition as ineffective and potentially damaging to students. Great thinkers like John Locke and John Dewey (among many others) claimed that children learned best by observation and personal experience, a pedagogical theory I absolutely agree with. The problem came about when the Progressives finally had the opportunity to explore their theories, once enacted the grand ideas of experiential learning quickly degraded into repetitive worksheets describing situations and asking how students would react to them. The great thinkers of the movement couldn't imbue all the teachers with the ability or resources to truly teach by doing, and they fell back into old habits with repeatable results. Most teachers cannot or will not teach a subject which has no right answers, and requires students to be comfortable with that.
The second piece of history Shammas leaves out is the No Child Left Behind act, which standardized curriculum and (more importantly) testing throughout the country. The reason philosophy can't be taught is the same reason he believes it will help people learn: there are no right answers. A federal standardized testing authority cannot judge whether a student has made a moral, empirically supported argument or a sloppy claim to genius; they have neither the funding nor the objective viewpoint to accomplish this.
This all leaves out the fact that if a student does not want to learn something, they won't. Attempting to introduce students who may already have dismissed education as impractical to their lives to Nietzsche and Kant is unlikely to derail their plans away from school.
None of this means that the subject is unworthy of public education, or that it's simply impossible to implement, but many other pieces need to fall into place before such a proposal could be appreciated.