An early history of Husker Thanksgivings: (1890-1950)
Independent
1890 Nov 27 - Omaha YMCA
1891 Nov 26 - Iowa
Western Interstate University Football Assoc.
1892 Nov 24 - Iowa
1893 Nov 30 - Iowa
1894 Nov 29 - Iowa
1895 Nov 28 - Iowa
1896 Nov 26 - Iowa (tie. played again on Saturday Nov 28)
1897 Nov 25 - Iowa
Independent
1898 Nov 24 - Iowa
1899 Nov 30 - Grinnell College
1900 Nov 29 - Minnesota
1901 Nov 28 - Haskell College
1902 Nov 27 - Northwestern
1903 Nov 26 - Illinois
1904 Nov 24 - Illinois
1905 Nov 30 - Illinois
1906 Nov 29 - Cincinnati
Missouri Valley Conference
1907 Nov 28 - Saint Louis
1908 Nov 26 - Wabash
1909 Nov 25 - Haskell College
1910 Nov 24 - Haskell College
1911 Nov 30 - (nobody) - first NU Thanksgiving with no football (played Mich Nov 25)
1912 Nov 28 - (nobody) - played Oklahoma Saturday before
1913 Nov 27 - (nobody) - played Iowa Saturday before
1914 Nov 26 - (nobody) - played Iowa Saturday before
1915 Nov 25 - (nobody) - played Iowa Saturday before
1916 Nov 30 - Notre Dame
1917 Nov 29 - Syracuse
1918 Nov 28 - Notre Dame
Independent
1919 Nov 27 - Syracuse
1920 Nov 25 - Washington State
Missouri Valley Conference
1921 Nov 24 - Colorado Ag (CSU)
1922 Nov 30 - Notre Dame
1923 Nov 29 - Kansas State
1924 Nov 27 - Oregon State
1925 Nov 26 - Notre Dame
1926 Nov 25 - Washington
1927 Nov 24 - New York University
Big 6
1928 Nov 29 - Kansas State
1929 Nov 28 - Iowa State
1930 Nov 27 - Kansas State
1931 Nov 26 - Pitt
1932 Nov 24 - Missouri
1933 Nov 30 - Oregon State
1934 Nov 29 - Kansas State
1935 Nov 28 - Oregon State
1936 Nov 26 - (nobody) - played Oregon State Saturday after
1937 Nov 25 - (nobody) - played Kansas State Saturday after
1938 Nov 24 - Kansas State
1939 Nov 23 - (nobody) - played Oklahoma Saturday after
1940 Nov 21 - (nobody) - played Iowa State Saturday after
1941 Nov 20 - (nobody) - played Iowa Saturday after
1942 Nov 26 - (nobody) - played Kansas State Saturday after
1943 Nov 25 - (nobody) - played Iowa Saturday after
1944 Nov 23 - (nobody) - played Kansas State Saturday after
1945 Nov 22 - (nobody) - played Iowa Saturday after
1946 Nov 28 - (nobody) - played UCLA Saturday after
1947 Nov 27 - (nobody) - played Oregon State Saturday after
Big 7
1948 Nov 25 - (nobody) - played Oregon State Saturday after
1949 Nov 24 - (nobody) - played Colorado Saturday before
In our early years, up until 1936, playing on Thanksgiving was definitely a Husker tradition, but the opponent varied. Often, it was a season-ending non-conference game. After 1950 we started playing Oklahoma more often as a season-ender, although not always on Thanksgiving.
I think it's fine to not have a regular opponent, as long as the Huskers keep playing on Thanksgiving or Black Friday. Kind of like the Cowboys or Lions, only a way older tradition.