I believe Arch died in Oklahoma, maybe Grady Co. where the family was living in 1930. Oklahoma doesn't publish their vital record dates which makes finding info more difficult. I looked through the indexed cemeteries and obituaries online, but didn't find him.
There was a Less Roach died June 9, 1930 and buried in Rose Hill Cemetery. I did notice on one family tree that his name was listed as L Archer Roach. Maybe him?
There was a Roach buried in the same cemetery with no first name listed. That person died May/June 1937 with the burial 1 June, block 12, lot 13 with a T. W. Roach b. 29 Dec 1851 and d. 30 May 1910.
You can request a copy of his death certificate from Oklahoma State Dept. Health, www.ok.gov/health/Birth_and_Death_Certificates/ but they have restrictions.
Joan Sunshine Roach
Born: 24 Mar 1926, Twitty, Wheeler, Texas
Father: L. A. Roach, 38, b. Corsicanna, Texas, farmer
Mother: Lola Pool, 32, b. Temple, Texas, housewife
Source: Texas Birth Certificates 1903-1935, familysearch.org
Joan Shumshire Roach
Born: 24 March 1926, Wheeler Co., TX
Father: L. A. Roach
Mother: Lola Pool
Source: Texas Birth Index 1903-1997
I think you mentioned that the entire family in 1930 was living in Chickasha, Grady, OK.
Lela Mae Pool Roach
Born: 12 July 1894, Texas
Died: 29 Jan 1936, Brown Co., TX
Source: Garvey/Owen/Crone/Bougher Family Tree, ancestry.com
Most larger public libraries provide free access to Ancestry with a library card. This tree includes many family members.
1935 Brownwood, Brown, Texas
1940 Justice precinct 1, navarro, Texas, State Orphans' Home
Gwendolyne Roach, b. TX, 10, pupil
Jauhanna Roach, b. TX, 13, pupil
Billie Jane Roach, b. TX, 11, pupil
Jauhaunna Roach Lewis
Born: March 24, 1925
Residence: Memphis 13 yrs
Occupation: Telephone operator
Died: July 28, 1956, Baptist Memorial Hospital, Memphis, Shelby, Tennessee
Husband: James T. Lewis
Father: Arch T. Roach
Mother: Lala Pool
Informant: Mrs. Paul Keltner, sister living in Memphis
Cause of death: acute pulmonary edema due to cerebral embolus due to rheumatic heart disease
Burial: July 30, 1956, Mt. Vernon Gardens Cemetery
Funeral Home: Cosmopolitan Funeral Home, 1900 Union
Source: Tennessee Death Records 1908-1958, ancestry.com. I can email this to you if you'd like a copy.
Jauhauna R. Lewis
Born: March 24, 1926
Died: July 28, 1956
Buried: Memphis Memory Gardens, Memphis, Shelby, TN
Source: findagrave.org and contains picture of her tombstone