Yes, the Smithsonian snubbed the Wrights. No, Wilbur did not die, "weakened by his nine-year dispute with the Smithsonian." He died of typhoid fever, If Wilbur was weakened by anything it was from the law suits he threw at anyone who tried to compete with him, especially Glen Curtiss. (It was during this trial that Curtiss re-worked the Langley design to make it negligibly airworthy.)
While their nation was slow to recognize them, the Wright brothers were fully celebrated in 1909, after Wilbur's triumphs in Europe the year before and Orville's claim of an Army contract in Washington. In June of 1909, the President of the United States personally presented them with a gold medal commissioned by the Congress.
The Smithonsonian and Langley's role in all this have been greatly exaggerated. Here for a fuller look at why Americans ignored the Wrights prior to 1909:
Early Automobiles and Airplanes: The Cultural Lag
For the record, from the Smithsonian itself (the Wright pages at the S was linked above):
http://www.nasm.si.edu/wrightbrothers/icon/feud.html
The roots of the rift between the Wrights and the Smithsonian began in 1910, when then Smithsonian Secretary Charles Walcott (Langley died in 1906) refused the Wrights’ offer of donation of the 1903 Wright Flyer, requesting instead a current Wright aircraft. Walcott intended to display the later Wright airplane with aeronautical artifacts of Langley, suggesting a connection between Langley’s work and the Wright achievement. The Wrights’ suspicions were aroused... Orville Wright’s concerns deepened in 1914 (Wilbur died in 1912) when the Smithsonian contracted aeronautical experimenter and aircraft manufacturer Glenn Curtiss to rebuild Langley’s unsuccessful 1903 full-size airplane, the Great Aerodrome, which crashed for the second time just nine days before the Wrights’ success at Kitty Hawk... After completely rebuilding the Langley Aerodrome with extensive modifications and a different engine, Curtiss did manage to make brief, straight-line hops with it. The aircraft was then returned to the Smithsonian, restored to its failed 1903 configuration, and displayed with a label stating that it was the “first man-carrying aeroplane in the history of the world capable of sustained free flight.” Orville was outraged.
http://www.nasm.si.edu/wrightbrothers/icon/feud_2.html
In the face of Orville’s action, the Smithsonian continued to dodge the issue. They offered only an unsatisfactory compromise on the language of its label accompanying the Langley airplane on public display, and did so, in the words of new Smithsonian Secretary Charles Abbot, “not in confession of error, but in a gesture of good will for the honor of America.” The comment only served to stiffen Orville’s resolve to gain satisfaction. Even Charles Lindbergh offered to help mediate the dispute.
http://www.nasm.si.edu/wrightbrothers/icon/feud_3.html
The Wright Flyer would remain in England until 1948. In its 1942 annual report, the Smithsonian finally published the article, entitled “The 1914 Tests of the Langley Aerodrome,” recanting its views on the Langley matter that Orville wanted. In 1943 he made plans to have the Flyer returned to the United States and transferred to the Smithsonian for public display... Orville died suddenly of a heart attack in January 1948 while the Wright Flyer was still in England, leaving it to the executors of his estate to fulfill his wishes and bring the treasured artifact home. It was installed at the Smithsonian in an elaborate ceremony on December 17, 1948, 45 years to the day after its history-making flights.