Just looking at the different business classes:
I woudld try to go on SA. They have flat/horizontal business class seats. LH, LX and many other still have the older style slopes.
I believe SA goes to Frankfurt, London and Munich from europe in addition to their direct flights to the US.
I amnot familiar with the continental program, but the US program allows only one stop, either inbound or outbound. Since traveling easy is typically harder with jet lag and you want to be more relaxed when on vacation i would try to go to europe first from the US to one of the points where SA flies to, stay a few days, then go to South Africa. Then take a flight directly to the US back.