Civil War soldier or not?

It's a truism that families preserve inaccurate traditions. They may contain an element of truth, sometimes they turn out to be false. One of my father's family traditions was that his great-grandfather David Wheeler White (1834-1879) had been a Union soldier in the American Civil War. There was also a David W. White who served in Company E of the 33rd Indiana Infantry. However, this man survived to receive a pension in 1883, whereas my David died in 1879.

So far I have no evidence that he served on either side. I know that his brother Will (William P. White, 1842-1924) served in Company C of the 116th Indiana Volunteers. Rosters of regiments composed of men from Fountain County, Indiana do not contain a David W. White.

According to the 1860 federal census my ancestor David W. White and his family were living in Vermilion County, Illinois, just across the state border. Just to complicate matters, there was indeed a David W. White from Vermilion County who served in the Union Army, but as far as I can determine this is another man, not my David. This second David W. White lived in Vermilion County after the Civil War, while my ancestor moved to Kansas by 1868.

I have searched Indiana, Illinois, Missouri, and Kansas rosters for my 2g-grandfather, both Union and Confederate. My 2g-grandmother Hannah White is not enumerated in the 1890 federal census of Civil War veterans and their widows (she died in 1907).

If David W. White did not serve, why not?

According to my great-aunt Anna Mae, David W. and Hannah White moved west on doctor's orders, in order to find a healthier climate for David. No source says it explicitly, but on the surface it sounds like my ancestor suffered from a long-term illness, perhaps tuberculosis. Perhaps that was the cause of his death in 1879 at the age of 44. Perhaps his illness prevented him from serving in the army.  In his 1853 letter to his brother David I mentioned in an earlier post, Alexander White (David W.'s father) makes a passing reference to his son David's illness but doesn't specify what it was.

Another possibility has to do with religion. In her autobiography, David W. White's daughter Abbie Willena Baker (1873-1968) relates that her father belonged to the Brethren church. I haven't yet determined if "Brethren" means the United Brethren in Christ, or if it refers to another group, the so-called Dunkards. There were United Brethren congregations in Fountain County in the nineteenth century, but I haven't yet found out if there were Dunkards. Perhaps David W. White did not serve in the army because of his religious beliefs.

As usual, one question leads to more questions.