"Franchising" was I believe to be done by the Interstate Commerce Commission and gave the designated bus company exclusive right to operate on a particular highway. When this began and what led to it, I do not know. Some of the early bus companies were railroad operated and began as a means to get rid of branch line passenger trains. Somewhere along the way most of these things were politically killed. I can give a couple examples which may or may not help. The one railroad owned bus company that lasted into the 1960's at least, was Gulf Transport Company owned by the Gulf Mobile and Ohio Railroad. This appears to have been set up to replace passenger trains on their routes. Most of their trains, post WW2 at latest and post the major campaign to get Mississippi's highways paved in the 1930's in general usually did not carry a bus load's worth of passengers, anyway. Somewhere buried away I have one of their system timetables from around 1960. In general their routes followed the routes of the pre-Alton takeover GM&O. The GM&O had two low density main lines in eastern Mississippi, with Jackson TN as their northern hub, plus several branches. My main knowledge relates to southeast of Memphis. Where Greyhound operated, they used Greyhound stations. Such was the case in Memphis. Most of their bus routes were a few a day and lightly loaded, primarily because of the low density of the population in the areas served. You say, "why into Memphis?" The GM&O didn't serve Memphis. Yes they did. They had trackage rights over the Southern out of Corinth MS. It appears that one of the reasons they lasted independently is because no one really wanted their routes.
An example of franchising: Between Memphis, Nashville, and Knoxville, Greyhound was apparently first to the table and held the right to operate on US 70 all the way. When Trainways wanted in the picture, they were given the right to operate on US 64, and then I think it was state highway 18 to Jackson TN and then state highways 20 and 100 to Nashville. (State 20 is now US 412.) East of Nashville their route was US 70N, which was not the original US 70. That route became US 70S. By the mid 1950's the practical run time was about the same on both routes. When the Interstates began to be open it appears that there was no franchising, but approach to them had to be via a road on which they did have a franchise. Thus, when I-40 opened for most of the Memphis - Jackson TN distance, Greyhound accessed it via US 70 on the Jackson TN end and Trailways accessed it via TN 20. (now US 412)