A cursory look at the list of American Commanders on the wiki page shows that the 12th president Zachary Taylor and the 14th president Franklin Pierce were commanders during the war. Furthermore, the 18th president Ulysses S. Grant served in it plus several failed candidates for president. That's excluding the fact that many of the commanders eventually served as territorial governors or representatives/senators in various states. Why did this war produce so many statesmen in mid-19th century America?
The Mexican-American war is often forgotten today in the shadows of the World Wars, but at the time it was a tremendously popular conflict. One of the major reasons behind this is that the Mexican-American war was the first American war to be covered extensively by war correspondents and newspapers, mainly out of New Orleans. The late Tom Reilly put together the source on the role of reporters in the war titled War with Mexico!: America's Reporters Cover the Battlefront. But war correspondents weren't the only media innovation introduced to report on the war, as new telegraph links across the country and the Associated Press both emerged to handle the huge public curiosity about the progress of the war.
It's also worth mentioning that many of the US soldiers sent were not professional soldiers, but rather military volunteers who joined with a remarkable enthusiasm. The incredible number of volunteers, a keenly developing national patriotism, and expanded reporting capabilities brought civilians into direct involvement with the goings-on of the war.
Another surprising feature of the Mexican War was the sheer lack of bad news from the American perspective. Mexico was a deeply fractured country during the 19th century, still recovering from the tremendous aftershocks of independence and dozens of ongoing conflicts within its borders. I covered aspects of this in a previous post. To quote:
At the height of the conflict in 1847-1848, more Mexican troops were fighting an internal civil war than were engaged in the war against Americans, northern Mexico had lost over 80% of its population, and an entire country seceded in the south. The government of Mexico, which we think of as a single entity was itself a mess of competing factions, more akin to a herd of cats than a coherent institution.
Although the US troops were better equipped and better supplied than their Mexican counterparts, they were astonishingly successful in their own right, achieving victory in nearly every engagement.
It was truly a perfect storm of factors: emerging national identity as 'Americans', a rapidly developing media network accessible to everyone, huge popular support, and an almost idyllic war for a public unused to the horrors they normally encompass.