That is incorrect.
The US Experimented with throttleable engine designs going back decades. The X-series were all experimental. That's what the X stands for.
The X-15 has nothing to do with carrier rockets into space anyway. It was a research program that probed the limits of aircraft speed and altitude given the technology of the day. It provided the US with invaluable flight data that was later incorporated into the SR-71 Blackbird.
The X-15, which is an airplane, did have a liquid throttleable re-startable rocket engine. It was dropped from the underbelly of a B-52 at high altitude in order to give the engine plenty of time to start. This is because the liquid throttleable engine still had ignition reliablility problems.This is not something that a carrier rocket could afford to have. As I said before, all the engines in the first stage of a carrier rocket needs to fire simultaneously, or the rocket will tear itself apart. The US did not solve this problem, the liquid rocket engine was discarded and the technology for liquid throttleable engines for use in carrier rockets was never developed. That is, not until the 1990's when the a US corporation bought the technology from Russians.
Which brings us to the discussion of the defining the limit of space.
According to the Federation Aeronautique Internationale, space is arbitrarily defined as the altitude above 100 km over the earth. By this arbitrary definition, the X-15 barely managed to make it into 'space' twice.
However, that altitude is far, far below the minimum altitude needed to maintain low earth orbit. It is of no significance whatsoever to a manned space program unless you severely limit yourself to sub-orbital flights only (which is absurd).