I have to disagree. It fits but it doesn't, first a school trained and certificated physician from one of the more prestigious schools in the country at the time likely would not have gone to the plains as a private to fight Indians in my opinion. To further cast doubt on what appears to be a fit is the original muster sheets of the 19th Kansas Cav from its 1868 organization that does list a Thomas Smith as a private Co L. as a 21 year old from Paola Kansas. I can't make that Thomas Smith fit Thomas Kerrins Smith

see: https://www.kansasmemory.gov/item/227708/page/107

Further reading in History of 19th Kansas does not mention Smith, whom we are trying to make notable because of his antecedents, and if truly the same I would be surprised nothing was said about him being the son of a Lt Gov from Missouri.

https://www.kansasmemory.gov/item/210758/page/1

I believe the Thomas Smith in the 19th Kansas is Thomas Alexander Smith born 1846 Platte Co Missouri found in 1860 in Johnson Co KS with parents:
Name T A Smith
Age 14
Birth Year abt 1846
Gender Male
Race White
Birth Place Missouri
Home in 1860 Oxford, Johnson, Kansas Territory
Post Office New Santa Fee
Dwelling Number 715
Family Number 815
Cannot Read, Write Y
Inferred Father
Alex Smith
Inferred Mother
H L G Smith
Household Members (Name) Age
Alex Smith 40
H L G Smith 32
T A Smith 14
L J Smith 12
W H Smith 8
Jno A Smith 6
J B Smith 5
G W Smith
M E Smith 1

As cool as it would be to have a nice Indian fighter story for Thomas K Smith I just am not convinced yet. There is just too much confounding evidence and I can't get to preponderance of evidence much less beyond reasonable doubt. :>)