I've never actually heard anyone call them feeder roads... a search of the Dallas Morning News archives reveals 8 hits for "feeder road" but 500+ for "service road" ... maybe its just a DFW nomenclature though. In the rural areas they are signed as Frontage Roads.
Jack Kultgen was head of TxDot during the early Interstate construction area and built Frontage Roads, as previously stated, to services large ranches and farms - but after 1968 they were built primarily at the behest of property developers like Rosewood or Wynne - at least in the DFW area. I dont know a ton about Houston.
TxDot stopped development and construction of Frontage Roads after 1991, unless plans were specifically in place and budgeted to build them in advance of a future freeway; or if an existing road was transformed into a freeway. For instance, SH 130, Loop 49, the Fort Bend Parkway, and other new limited-access roads generally don't have them. SH 121, 160, Marsha Sharp, etc, have them as they're upgrades of existing roads. SH 190 was built as frontage roads because TxDot was not allowed to build toll roads at the time, so the main lanes were to be managed by NTTA.
Other things TxDot decided in 1991: to change all ranch roads into farm roads, and to remove all cloverleafs in the state. Those two policies are still hit and miss.... (cloverleafs are rapidly dying here, though)
Again, my personal knowledge does not generally extend to Houston.... I need to spend a weekend there.