To back up that theory, I've read through the tax lists of Pulaski Co., and discovered that the younger John there left Pulaski Co., KY at about the same time your John appeared up in Indiana.  This John, as you may recall, had married Rachel Price on 9 May 1822.  Well, my big bit of information this time is that the widow Rachel Brinson married again in 1850 -- up in Shelby Co., Indiana!  This really solidifies the link back to Pulaski Co., Kentucky (which mainly consisted of handwaving on my part, I'll confess), plus explains why your John was so scarce in 1850 -- he had died.

I found this out last week, when I had the chance to spend a few hours in the libraries in the Greensburg (Decatur Co.) and Shelbyville (Shelby Co.)

Here's what I think I know about the family of John BRINSON and Rachel PRICE:
John BRINSON, born about 1802 in Pulaski Co., KY, died between 1840 and 1850 in Shelby Co., IN.  He was the son of John BRINSON and Belinda TAYLOR.  Moved from Pulaski Co., KY to Decatur Co., IN just before 1830, and was counted in the 1830 census there.  Counted in the 1840 census in Shelby Co., IN.  Married in Pulaski Co., KY on 9 May 1822 to Rachel PRICE, permission to marry given by John BRINSON Sr.

In May 1831, five lots of corn were sold at O. WELSH's by William CRAWFORD, the administrator of Lewis CRAIG, to himself, James ALEXANDER, John BRINSON and William LOCKRIDGE.  Decatur Co., IN court records.

Also in 1831, John BRINSON of Clay Twp., Decatur Co., IN, registered the following stock mark: a crop and slit in the left ear, and the upper bit out of the right ear.

19 Dec 1839 - mentioned in the estray records for Sugar Creek Township, Shelby Co., IN.

Rachel PRICE, after John died, was married 12 May 1850 in Shelby Co. to John DERICKSON. She died 19 Feb 1866, aged 64y, 10m, 10d, and is buried in Shelby Co., IN at the Little Blue River Baptist Cemetery. In the 1850 census, she and John DERRICKSON were counted in Union Twp., Shelby Co. He was aged 61, born in NJ, and she was aged 46, born in KY.  They're also in the 1860 census, for Shelby Co. on page 81-536.  The index gave her as 57, born KY, and John DERRICKSON as 71, born NJ.

Children:
James H. BRINSON
Was he a teamster in the Mexican War, who applied for a pension in 1889 while living in IL?

He married first Susan PIERSON, the daughter of Robert and Nancy PIERSON.  She is buried in Van Pelt Cemetery, and her stone says she was born 1 Aug 1834, and died in 1854.  She and James H. BRINSON were married in Shelby Co. on 15 Jan 1853.

They had Telitha J. BRINSON.  She married Jacob C. CRAWFORD in Shelby Co. on 28 June 1875.  Telitha was born in about February of 1854, in Shelby Co., IN

He married second Virginia ALEXANDER in Shelby Co. on 4 Oct 1857.

Dec 1854 - James H. BRINSON, guardian of Telitha Jane, the minor heir of Susan BRINSON, sold real estate to Green B. McCARTY.  Telitha's grandmother is Jane SHELBY, formerly Jane PEARSON, widow of Robert PEARSON.

Sarah BRINSON, born about 1827 in KY, married Joseph PIERSON in Shelby Co. on 29 Dec 1843.  He was born about 1823 in KY.  They had children Mary E. (born about 1845, Shelby Co., IN) and James A. (born in about July 1849 in Shelby Co., IN).  Sarah and Joseph PIERSON counted in the 1850 Shelby Co., IN census.

Willis G. BRINSON, born about 1824 in KY, married Matilda.  She was born about 1830 in IN.  Willis G. is buried in the Brandywine Cemetery in Shelby Co., where his stone says he died 3 Apr 1852 at the age of 29 years.  After Willis death, Matilda married John BRADY in Shelby Co. on 18 Apr 1854.  The 1860 Shelby Co. census index states that Matilda and John BRADY were on page 213-708, that both were aged 30, and born in IN.

They had
1. Lynn Boyd.  He was born in about July 1849 in Shelby Co., IN.  Buried in the City Cemetery, Shelbyville, his stone reads 1849-1908.  His wife was Eliza H., who is buried in the same cemetery, as are two of their infants.  Her stone reads 18 Feb 1852 - 3 Jan 1880 or 1888.  He apparently had other wives, as Shelby Co. records show marriages on 20 Jun 1872 for Lynn B. BRINSON and and Lida H. MARTZ, and on 14 Aug 1881 for Lynn B. BRINSON and Fannie T. CRAWFORD.  The Volunteer (a newspaper) of 8 Jan 1880 says that Lydia, the wife of L.B. BRINSON "of Sabetha, KS" died on Saturday, that she was the daughter of Joseph L. MARTZ, and that she was to be buried in Shelbyville, so presumably Eliza H. and Lydia are one and the same, and someone incorrectly transcribed the tombstone.

2. Ardenia Jane, born about 1851 in Shelby Co., IN.  She is buried in the Brandywine Cemetery (as is her father).  Her stone says she died 30 Mar 1862, aged 11 years.

8 Aug 1850 - Willis G. BRINSON bought real estate in Shelby Co. from Elias BASS, the guardian of David T. BOON.

11 May 1852 - Coleman STORMS appointed the administrator of the estate of Willis G. BRINSON.

4 Jan 1853 - James F. RULE appointed guardian of Lynn Boyd and Ardena Jane, heirs of Willis G. BRINSON.

16 June 1854 - Apollos KINSLEY appointed guardian of Lynn Boyd and Ardena Jane, heirs of Willis G. BRINSON.

Sept 1854 - Matilda BRINSON, mother of Lynn B. (aged 5) and Ardena J. (aged 3) sold real estate formerly owned by Sebastion BARTH.

9 Sept 1854 - Apollos KINSLEY, as guardian, sold real estate to Matthew WILLIAMS.

4 Mar 1861 - Apollos KINSLEY resigned as guardian.

8 Mar 1861 - John A. WHITED appointed guardian.

Frank M. BRINSON

Co. H, Reg. 16, in 4-13-1861, out 5-23-1862, served from Shelby Co., IN.