The 1960s would be too late for any subtle, natural-looking TLs. Americans were already tuning into football and basketball in impressive numbers, and baseball has been the king for a century.
Your best bet might be to have a very long string of gruesome injuries in televized football games. If football can be seen as just too brutal, it might suffer the same baklash that boxing and animal-fighting faced around that time as America's values changed, and either become a minor sport (boxing) or illegal (animal-fighting). The problem is, in OTL 1960s, it was hockey that was the just-a-bit-too-violent sport, whereas football was finally getting its act together with proper helmets and stricter penalties. So you'd also have to reform hockey ITTL, or else it won't take advantage of football's image problem.