As most historians have noted, Theodore Roosevelt did lead his soldiers into a trap and suffered the most causalities during the Battle of Las Guasimas and the Battle of San Juan Hill. It has been noted that he acted recklessly while leading his group into battle.
Because they were so damn cool.
The Rough Riders were all volunteers who signed up out of a sense of adventure or patriotism. Wikipedia says that most of them were "college athletes, cowboys, and ranchers." Roosevelt was a privileged aristocrat and Assistant Secretary of the Navy. But when war broke out (a war he pushed for), Roosevelt resigned his cushy post and helped form a rag-tag group of soldiers trained for combat.
If "it has been noted that he acted recklessly" by leading the charge up Kettle Hill, it certainly wasn't by his contemporaries. Roosevelt was honored with the Medal of Honor. His citation noted that the Rough Riders calvary charge changed the entire course of the battle.
Lieutenant Colonel Theodore Roosevelt distinguished himself by acts of bravery on 1 July, 1898, near Santiago de Cuba, Republic of Cuba, while leading a daring charge up San Juan Hill. Lieutenant Colonel Roosevelt, in total disregard for his personal safety, and accompanied by only four or five men, led a desperate and gallant charge up San Juan Hill, encouraging his troops to continue the assault through withering enemy fire over open countryside. Facing the enemy's heavy fire, he displayed extraordinary bravery throughout the charge, and was the first to reach the enemy trenches, where he quickly killed one of the enemy with his pistol, allowing his men to continue the assault. His leadership and valor turned the tide in the Battle for San Juan Hill. Lieutenant Colonel Roosevelt's extraordinary heroism and devotion to duty are in keeping with the highest traditions of military service and reflect great credit upon himself, his unit, and the United States Army.
I would add that "acting recklessly" is sometimes appropriate or praiseworthy in military situations. I don't doubt that Audie Murphy acted recklessly when he ordered his men to fall back, jumped onto a burning tank destroyer, stayed there for hours, and single-handedly killed 50 or so advancing Germans. But his daring and sacrifice is seen as the exemplar of heroism, and Murphy is considered the greatest American soldier to ever live.
In conclusion, suppose that Dick Cheney had resigned his post after the Iraq War broke out and traveled to his home state of Wyoming. There, he rallied a group of Indians, ranchers, cowboys, college students, veterans, and eccentrics. His All-American Gang lands in Iraq. During the critical battle of Fallujah, Cheney personally leads a suicide charge through the streets-- on horseback!-- turning the tide of the battle and allowing America to win a crucial victory.
Hell, I'd vote for him.
Short answer(s). Because he won. That is, his side won the battle, thanks in large part to his efforts, and his country won the war. His charge of San Juan Hill was seen as one of the high points of an otherwise "dull" war.
And "history is written by the victors."