Nebraska didn't require the recording of deaths until 1904, so it is doubtful the courthouse would have that information.
Newspapers are the best resource for deaths during that time period.I found death notices for Rev. Betts in Falls City's two newspapers.They both give his death as occurring on July 2, 1878. The Globe-Journal of Sat. July 6, 1878 (editor Jacob Baily) has under "Rulo Items" - "Died-On Tuesday July 2d 1878, the Rev. Thomas Betts." The Globe-Journal of July 13 under "Rulo Items" mentions that "Rev. George Betts, son of the late Rev. Thomas Betts is in the city."
The Falls City Press of Thursday July 7, 1878 (editor Andrew J. Reed) printed this announcement: "Father Betts is Dead!" "The above announcement from Rulo shocked this entire community yesterday morning.He died on Tuesday evening at his home in Rulo.We have no particulars, but know that he has been quite feeble for the past two weeks.A week ago yesterday he wrote us as follows:My Dear Mr. Reed: - I am so poorly that I do not hope to hold services in either church on Sunday,Yours sincerely, Thomas Betts. We have heard that the funeral would take place today.A good man has gone to his reward."
The Falls City Press of Thursday July 11 printed an obituary of Thomas.This paper also printed a letter from Thomas' son George C. Betts that gave a biography.It gives details of his early life in Ireland.This biography consists of almost a whole newspaper column.