I think we would have a better, stronger society than we do now.
Suppose you're right. LBJ ends up becoming President anyway in this alternate timeline. We know that he'll propose the Great Society programs which ended up destroying the Black family in America.
I don't know how much you know about LBJ, but he was a deceitful conniving bastard.
Unpacking alternate timelines in history can be tricky, especially if they are a long distance in time from the alternate branch point, but we only got LBJ as a result of Kennedy picking him.
We only got Kennedy through vote fraud in Chicago, and because Joe Kennedy had become wealthy and powerful as a result of his criminal doings, mostly bootlegging, but also extortion and such.
How did alcohol prohibition occur? Liberal women "progressives" pushing for the 18th amendment. (and the 19th.)
Now that i've gotten to prohibition, I want to take a little side branch here. A couple of weeks ago I watched a video featuring Jay Leno teaching a young man how to drive a model T. During the video, Jay Leno mentions that Henry Ford used to go on camping trips with John Rockefeller and other wealthy men of that era. Ford had mentioned to the group that he had designed the Model T to run on Alcohol, because he felt that he would have a good market in selling the vehicle to farmers, who could distill their own grains into fuel for the car.
Rockefeller then started financing the Prohibition movement, which would have made sure alcohol would never be a threat to his standard oil company.
Now back to the branch timeline.
The progressive era featured Northern women activists from Republican families, meddling about in politics and getting social changes enacted. There is a good chance that without the civil war, these women would have never attained the levels of power and influence that they came to have.
No prohibition, no Joe Kennedy, no John Kennedy, and no LBJ, and therefore no "Great Society."
If nothing else, I hope this view entertained you.