It's not merely that the South was Solid. It's also that the South was simply not strong enough within the Democratic coalition to get the nomination and the South itself was too different from the rest of the country to build a proper coalition within the Democratic party needed to do this.
It was conventional wisdom back before the 1960s that a Southern "regional" candidate had no chance at a Democratic convention, which by sheer force of demographics would be dominated by non-Southern delegates, including many representing urban, leftist and minority constituencies within the party. And the very qualities and political views that made Southern politicians sufficiently prominent in their home states defined them as, at best, regional Southern candidates.
There is a wonderful discussion of this in the 3rd volume of the LBJ biography by Caro (I presume, even more is there in the 4th volume, but I haven't yet read that one). LBJ had to really find a very fine ballance between not alienating his own Texan electorate, maintaining support among the other Southern politicians, and getting any support outside of the South.
Ending segregation and enfranchising Southern blacks made it possible for a Southern politician to be successful in his home state without being a fire-breathing segregationist - which, in turn, made them fair presidential material. Emergence of GOP to take the more right-wing electorate in the South also moved the median of the Dem electorate in the South closer to that of the rest of the party. Note, that the 4 successful Southern Dem nominees (LBJ in 1964, Carter, Clinton and Gore) were running from fairly liberal positions on matters such as race - there was no way such positions could have been sustained by successful Southern politicians before the 1960s (Gore Sr. being a partial exception - but then TN is not fully Southern in any case). The Republican Bushes, on the other hand, are barely "Southern" culturally.
Well Texas is so overrun by transplants these days (many if not most of whom are quite conservative) that it's not too surprising that the Bushes would still have strong Southern appeal.
