Their ordinary walking is red nucleus walking. Remarkably, this is true even for ground-walking primates such as rhesus monkeys. Yes they do, and the settings are clinically important. Those babies will crawl, a form of red nucleus walking, until the corticospinal tracts mature. At that point, they can learn to stand up and walk with two instead of four limbs.
Some babies never crawl, but stand up and walk as their first voluntary locomotion.
Presumably, in such babies, the corticospinal tracts myelinate at the same time as the rubrospinal tracts. You will find that for you and for normal passers-by, the arms swing involuntarily, in such a manner that as a given leg is extended the ipsilateral arm is flexed. Suppressing this arm-swinging takes thought and conscious effort. If it is absent, something is wrong. This may be an indicator of CNS abnormality, notably Parkinson's disease.