Plausibility check: X-15 being the first spacecraft to carry human in space, preferably in 1960-1961.
Your problem is at the time international standard said space was at LEAST 100km altitude, (the "Karman line") and while the X-15 could hit it note that it only did so twice, (flight 145 and 149, officially 90 and 91) the pull-out and aerodynamics were ROUGH. (
https://en.wikipedia.org/wiki/X-15_Flight_90,
https://en.wikipedia.org/wiki/X-15_Flight_91) Hitting the proper speed isn't difficult, (around Mach-6 for the X-15, a bit over Mach-4 for the lighter Spaceship One) coming down on such a high angle trajectory makes for high G's and high aerodynamic stress.
Also note that the X-15 (
https://en.wikipedia.org/wiki/North_American_X-15) eventually included altitude as a test parameter it was not in the original charter. The X-15 was originally to be a high supersonic/low hyper-sonic (over Mach-5) test aircraft to study the effects and problems with high speed atmospheric flight. In fact the other contractors that put forth ideas for the X-15 contract were pretty much down-selected BECAUSE their vehicles had additional altitude and near-space components that both the Air Force and NACA, (at the time, and after the Navy dropped out) were specifically NOT interested in. The high altitude requirements were added back in only after NAA got the contract.
In fact it turned out the X-15 as built could not in fact deal well with the strain of either very high altitude flight or very fast (over Mach-6 which was an original goal) flight.
In fact the US could have beaten the USSR into 'space' with Mercury since the original plan called for Alan Sheppard's flight to take place in early March but issue during the flight of "Ham" on MR-2 caused Von Braun to have doubts about a manned launch. Despite his usual style of getting opinions and gaining consensus before 'deciding' a course of action he flat out refused to budge despite numerous engineers and even the astronauts insisting he go ahead. So the booster for the original flight was used on the unmanned MR-1A and they had to wait for a new booster to arrive before preparations began. And it would have been an American 'first' straight up since it greatly exceeded the international definition of "space" and even though "only" suborbital it would have been a fully visible operation unlike Gagarin's flight. (Yes he'd have still been the first man in orbit but they would have NOT contested or disparaged the US "first" since doing so would have brought criticism on THEIR operations which were not as transparent. And the LAST thing they wanted was outsiders asking questions since they did in fact violate the agreements over "record" flights with Gagarin not landing in his spacecraft)
And that was the main problem with why the "Orbital" X-15, (actually 'just' short of a full orbit due to being a lifting spacecraft and therefore dropping into the atmosphere earlier than a ballistic vehicle) since no amount of rebuilding was going to make the X-15 air-frame able to survive reentry AND make a controlled landing. So the idea of having flown the X-15 into space probably would have no effect on the X-20 development since it had major political, mission, operational and design issues all on it's own.
For those who hate on McNamara let me just point out that the military had, (as they had in 1948) NOT expected the person who actually won the Presidency to do so and so started a number of very expensive and only vaguely justified programs with the expectation they would not be called to justify them to much and the X-20 was one such program. Specifically the US Air Force had removed any and all cooperative elements to the development and funding for the "X-20" (which was originally pitched and supported as a high-hyper-sonic test aircraft to go into the regions between Mach-7 and orbital speed) so as to cut the new "NASA" out of the program. Since NASA was only interested at the time in funding a 'test' aircraft and the military wanted an operational vehicle this was done on purpose. It is little wonder that NASA declined to co-sponsor the development after that and the Air Force was quite clear that even if NASA had been willing they would have had little or no input beyond some scraps. (As it was the Air Force still used NASA facilities and personnel but then forbid either the release of data or involved personnel to discuss any of the operations which was directly against NASA's mandate) Worse from the Administrations point-of-view the Air Force made it clear that the vehicle would be operated as and only available to the military, (Air Force) and would neither cooperate with or include NASA or civilian management in the planning or operations of that vehicle. This clearly would be a difficult issue in any talks with the USSR but the Air Force clearly did not care.
If you think that's NOT of great concern to civilian politicians then you clearly miss where that same Air Force was VERY interested in pushing for the construction and operation of a "fleet" of Orion propelled "Battleships" (
http://www.thespacereview.com/article/2714/1) carrying hundreds of nuclear warhead and being propelled by thousands more. It's simply not a credible scenario since at a minimum the USSR has to do the same OR they will have to strike first before those battleships are operational, there is really no other choices available.
(As an aside I've actually put together a scenario where McNamara actually tries to convince Kennedy to allow the Air Force to do just that and his reasons even make sense though his plan it to only allow a few to be built and operated and NOT as 'battleships'. In the end the Air Force is forced to work with NASA, first on the X-20 and then on a more capable follow on to support those Orion's which if rather fitting I think)
I assume that as the X-15 was a USAF program and the government wanted a "civilian" space program, the X-15 would have to be transferred to NASA if it was going to be developed as a space-capable vehicle. Come to think of it I recall that NAA had proposed an orbital X-15 but I don't know how feasible it would have been.
Other way around the X-15 program was in fact a joint program between the Air Force and NACA/NASA which was only given approval BECAUSE it was a joint program. Eisenhower had made it clear that no space project with ONLY military support and backing, (barring the reconnaissance satellite program but that was not part of the 'main-stream' military which is why the official Air Force spy-sat program was canceled) was going to be approved and since the X-15 could fly into space, (being defined as the US or US military definition or over 50 miles high) then it would require NASA support. The Navy backed out before the contract was awarded but the Air Force was in it for the high speed data for future hyper-sonic bombers whereas NASA wanted high speed aircraft data and maybe some altitude work.
The NAA "orbital" X-15 is a confusing proposition as it was initially suggested before Mercury was chosen as the US manned orbital effort. In effect NACA/NASA had already decided on a ballistic capsule and pretty much rejected anything that wasn't one, NAA was not aware of this fact and proposed initially a 'beefed-up' X-15, (which hadn't even been built at the time) that would delete all the 'non-essential' equipment (like landing gear for example) and added heavier heat resistant materials to the wings and fuselage, (some ablative was used just as in later high speed X-15 flights but at the time it was not clear if the ablative materials would be able to keep the needed aerodynamics for higher speeds, testing showed it didn't which is why it was only used sparingly on the later flights) and launched using several stored Navajo boosters and finally either a cluster of Titans or maybe a Saturn-1 to boost it for an 'orbital' flight. Even then it would be a "once-around" flight with the pilot bailing out over the Gulf of Mexico. As noted above this means it would NOT in fact qualify as a 'manned-orbital-flight' and the Air Force said so and rejected the idea. (Stormy learned to listen to the REAL powers behind the completions after that and worked directly with Maxime Faget to design and "win" the Apollo Command Module contract) The delta-wing X-15 was in fact an attempt to keep the X-15 program going with a "new" X-15 design that could reach speeds in excess of Mach-10 but this was a threat to the X-20 so the Air Force did not support the design or the program.
I find it interesting that the highest altitude flights were about Mach 5-5.5 but the Mach 6+ flights were at about 1/3 of this altitude.
"Highest" had to boost pretty much straight up therefore the final velocity was limited by gravity, the high speed flights had a flatter trajectory for that reason and since the entire program is based on how air-frames handle in the atmosphere at high speed...
Randy