Can software really convert sheet music into an audio recording as well as a skilled human with a physical instrument and a microphone (as judged by the audience)?
"Converting sheet music into an audio recording" consists of two steps: recreating instrument sound and recreating artist expression.
Recreating instrument sound
Virtual instruments do exist and are widely in use. You tagged your question as synthesizer but while modeling sounds of real instruments was explored in the past, presently most instruments emulating sounds of real acoustic instruments base on sampling. Recording technique is developed since over 100 years, and today it works just perfect – you record a note played on an instrument, and then you play it back. It can't go wrong.
Typically an instrument consists of several or even several tens of GB of audio samples. The challenge, besides the obvious one of getting a consistent and high quality recording of all notes of the instrument, is to collect samples representing various dynamic levels and articulations. Another difficulty is to represent various nuances, like moving from one note to another, repeating the same note, ending a note, additional mechanical sounds made by the instrument...
Imitation is the most successful in music styles that don't rely too heavily on subtle instrument articulation changes, though the instrument producers keep pushing these boundaries farther.
Recreating artist expression
This is more difficult, as it's not well defined in technical terms. Sheet music represents certain idea, but the composer must rely on the musician's ability to interpret it. A musician will play each note differently. They will emphasize metric accents and phrases using capabilities of their instrument. A crescendo mark represents idea of increasing the volume, but in practice a simple gradual change may not create appropriate artistic effect. And "appropriate artistic effect" is a highly subjective term.
Various instrument offer various expressive capabilities. The most universal ones are timing and changes of tempo. Most instruments also can be played with various dynamics and may also provide various articulations. These nuances typically are not, or even cannot be precisely notated in a sheet music meant for humans. A music score fed directly into a virtual instrument, and thus devoid of these nuances will sound rather mechanical and not interesting.
Much better results can be obtained by converting sheet music to a MIDI sequence with manual adjustment of dynamics, timing and articulation of each note. It is a tedious process, and most importantly: manual. While some software tools can aid in it, human involvement is necessary. And recreating the expression of a skilled musician this way is still very challenging.
In practice MIDI sequences which are played by virtual instruments are often recorded by human musicians on MIDI controllers that capture their expression. This can yield very good results, but as a musician is involved, this doesn't fulfill the requirement of "converting sheet music into an audio recording by software".
Use of programmed music sequences emulating the real instruments is therefore the most successful in styles and situations that don't rely heavily on artist expression and interpretation, that have fixed tempo, or where said instrument is in the background. But as soon as we focus on artistic interpretation, a machine cannot replace a skilled musician. At least not yet.