This 500-page explainer of Barack Obama ’s as yet unfinished tenure in the Oval Office begins with a kind of warning label: “A president’s legacy takes years, even decades, to fully reveal itself.” That’s not the only reason why the decision to publish a book about Mr. Obama at this particular juncture might seem imprudent. On the one hand, the president is not even halfway through his second term; on the other hand, as the midterm election results made emphatically clear, much of America already seems sick of the guy. As someone who attempted a similar feat with Mr. Obama’s predecessor, I can attest that an author writing the first draft of an unpopular sitting president’s history will encounter indifference unless he or she brings something entirely new to the discussion.

Chuck Todd is well positioned to do just that. From day one, he has viewed the Obama White House at notable proximity: first as NBC’s chief White House correspondent and now as the host of the network’s venerable (if ratings-beleaguered) Sunday morning show, “Meet the Press.” Mr. Todd is the rare Washington commentator whose on-air musings are consistently backed by authoritative reporting. Before he became a TV personality in 2007, Mr. Todd was a standout political reporter for National Journal. He knows his subject, and he knows how to write.

Still, as the book’s title indicates, Mr. Todd does not have the wind at his back. Six years in, our 44th president remains a cipher, someone “everyone is still trying to get to know,” as Mr. Todd writes. Particularly vexing is a basic irony that this book seeks to address: “How does one of the most successful politicians in American history . . . appear to be so bad at practicing the basics of politics in the backrooms of Washington, whether on Capitol Hill, on K Street, or at the Pentagon?”

With an informal, digressive writing style, Mr. Todd gamely sets out to produce an answer to that perhaps unanswerable question. “The Stranger” should not be confused with the deeply reported books already written on the president by the likes of David Remnick, Bob Woodward and Jonathan Alter. Mr. Todd seems to have gathered his material predominantly for use in his on-the-air news analyses, with the result that his interviews are driven less by the authorial desire to psychoanalyze than the pundit’s need to respond to the topic du jour. That said, he has harvested a bounty of fascinating anecdotes as the fruit of his access to the Obama White House. We learn that the administration’s greatest achievement to date, the Affordable Care Act, originated from a throwaway line in a 2007 campaign speech. There was intense warring both within the White House and with liberal House Democrats over the heavily earmarked stimulus bill—and whether vetoing it would cost the president Democratic votes on the aforementioned health-care bill. While meeting in the White House with several Log Cabin Republicans to determine whether they had persuaded any GOP senators to vote for the repeal of the “Don’t Ask, Don’t Tell” military policy on gays, Mr. Obama grew impatient at their vague promises and snapped, “Collectively, you need one Republican.” The author offers up the amusing image of Presidents Clinton and Obama, shortly after the 2010 midterm election, trying to figure out how to get into the White House briefing room so as to stage their impromptu joint press conference about the president’s decision to extend the Bush tax cuts. (Mr. Clinton to press secretary Robert Gibbs: “Yeah, can you help us unlock it?”) And, following the killing of Osama bin Laden, Mr. Todd has Secretary of Defense Robert Gates helpfully suggesting a communications strategy to White House advisers: “It’s called shut the f— up.”

You’ll notice that none of these nuggets gets us into the psyche of the Stranger in Chief. Mr. Todd does offer up some tantalizing thoughts on the subject. He tells us that Mr. Obama’s former colleagues in the Senate believe that things had always come too easily for the one-term Illinois senator—and that for this reason they prefer to deal with Vice President Joe Biden , a war-scarred politico who can “actually understand their frustrations.” (The most revealing passages in “The Stranger” concern Mr. Obama’s more transparent vice president, whose affability is so reflexive that when the White House patched him into the phone of a different senator than the one he intended to speak with, he simply made a new friend. Mr. Biden, Mr. Todd reports, finds his boss to be thin-skinned.) On the House side, the author observes that Mr. Obama’s relationship with former House Speaker Nancy Pelosi suffered because he “had few ties to Pelosi”—which of course raises the question: Then why didn’t the nation’s top Democrat set out to develop some ties? Mr. Todd says that it has been easier for Mr. Obama to be “playing hardball with members of his own party” than with Republicans and that “nothing irks Mr. Obama more than the idea that he’s somehow a leftist or liberal; he believes that most of his ideas are old Republican ideas from another era.” Yet none of this is thoroughly explored in the book, and what we’re left with is an amorphous composite sketch of an undefined subject.

Mr. Todd has been busily demystifying election cycles at NBC, and in places “The Stranger” reads as if it were written in haste. Phrases like “a tad” and “thick as thieves” are repeated on the same page; inside-baseball knowledge on the part of the reader is often assumed. Jim Messina, the White House deputy chief of staff, emerges out of nowhere as Mr. Obama’s “point man” on the “Don’t Ask, Don’t Tell” repeal without any elucidation of Mr. Messina’s crucial role in the administration or how he got to the White House to begin with. Similarly, a set piece on gay former Republican operative Ken Mehlman ’s role in the same legislative effort fails to mention that Mr. Mehlman had been George W. Bush’s political director when Mr. Bush came out in favor of a constitutional amendment to ban gay marriage. Reference is made to “the Tea Party politics of the Obama era” without explaining to the reader what the Tea Party is or how it arose.

Readers of “The Stranger” unquestionably benefit from Mr. Todd’s Beltway-insider status, though at times I wish he had played this down. (At one point, he refers to White House chief of staff Denis McDonough as “Denis”; at another, he describes the exclusivity of the annual Gridiron Club dinner and then adds a footnote: “Full disclosure: this author is a member of the Gridiron Club.”) The closeness can blind him to the significance of his reporting. Mr. Todd notes, for example, that Republican House Speaker John Boehner did not bring the Senate’s 2013 immigration-reform measure to the floor because it had passed in the upper chamber by a comfortable majority but not a vast one, which is what would have been required “for Boehner to sell to the 100 or so [House] Republicans who might have been persuadable on this issue.” The dismaying starkness of that political reality would seem to deserve more than passing commentary. Meanwhile, the author mentions, but does not highlight, the brazenness of two White House political warriors—Stephanie Cutter and David Plouffe —helping to craft the president’s 2012 jobs legislation, an un-passable bill that would serve as “something to run on.” Mr. Todd labels the bill “largely symbolic.” Nakedly political is more like it. In such instances, “The Stranger” feels like the first draft of what is not quite the great book it could have been.

To that nagging question about how someone so intellectually advanced could be so politically stunted, the author’s final answer is that Barack Obama’s “arrogance got the better of him.” As a novel twist on this familiar theme, Mr. Todd theorizes that Mr. Obama’s happy experience as editor of the Harvard Law Review “gave him a false self-confidence that burns in him—and burns him—to this day.” At the same time, the author seems to be suggesting that where Mr. Obama’s cockiness ends an overabundance of caution takes over—especially true when dealing with any issue like guns, gays or race that might offend working-class voters. In such instances, Mr. Todd reports that Mr. Obama insists to the fretful idealists in his administration, “I’m enough change.” This rather remarkable assertion by our first black president comes off as simultaneously haughty and defensive. It may also be entirely accurate.