Has Science observed and recorded a stars birth?

The star formation process from giant molecular cloud to unobscured protostar is thought to take about a million years.

So the answer is no.

Similarly, there are very few large scale physical processes that occur in the universe on a human timescale. Nevertheless we are sophisticated enough to understand that you do not necessarily have to see something happening to know that it has occurred and work out how it happened.

Or are you asking whether all the separate phases of the star formation process have been observed in different places? The answer to that is broadly yes. The rarest (shortest) phases is the initial collapse to a "core" that is embedded within an obscuring molecular cloud. Nevertheless, such objects can be seen at sub-mm and radio wavelengths.