Suppose that for some reason McNamara takes a shine to the X-20 when he's appointed Secretary of Defense and keeps the program going instead of cancelling it in favor of MOL. No other changes in his behavior, he just favors one Air Force human spaceflight program over another. What happens?
It seems fair to say, given the record of a number of other McNamara-favored programs, that the X-20 will not be seriously threatened until the Secretary of Defense retires, which will most likely be in 1968 as per OTL. By this point, according to the intended schedule, the vehicle would have been launched into space several times, and in fact would be nearing the end of its planned flight sequence. Even assuming delays, it should be well along in its flight program by this time, so the Air Force would have somewhat successfully established its own human spaceflight program.
Frankly, I can't see the program continuing past the intended flight sequence. Assuming that the design worked, which AFAICT it would have, then it would have been a very successful experimental aircraft but not much of a spacecraft, and the relatively small and unimportant Air Force HSF community, while certainly enlarged and emboldened by the program, will probably not be able to get the funding to start a big expensive new program for a larger, more militarily useful variant when orbital bombers are out of vogue and spy satellites are really starting to strut their stuff. The results will likely flow into the larger shuttle program that was starting to gear up about 1968 in terms of aerodynamic design and materials, but even then it cannot demonstrate many of the most important failures of the shuttle IOTL, so it is likely that NASA will make many of the same mistakes (in particular, it won't show that silica tiles require so much refurbishment, since they hadn't been invented yet!). Unless it, say, led Hubert Humphery to be elected and continue the space program, I can see it becoming something of a footnote to the larger space program, in much the same way as the X-15 or the HL-10, M-2, and X-24.