A historian refers to The Rough Riders, under Roosevelt's command take San Juan Hill. It is well known that the hill they actually took was Kettle Hill.
A quote is given saying, "The vice presidency is worth a bucket of warm spit." The actual quote from John Nance Garner was, "The vice presidency is not worth a bucket of warm spit," and that quote wasn't made for 30 years into the future. Therefore, it was irrelevant to Roosevelt becoming vice president.
The program depicts Theodore Roosevelt being at the top of a mountain when he is informed of the shooting of President McKinley. In fact, Roosevelt was in the home of the Lt. Governor of Vermont preparing to make a speech when he was informed.
The Theodore Roosevelt in the opening scene preparing to walk out of the hotel to the attempted assassination places a pen on the table having been editing his speech writing with his left hand. 20 minutes later, the Theodore Roosevelt beginning his political career in NYC is seen writing with his right hand.