Does leaving some of the Sunnah acts of prayer—such as reciting aloud when praying alone—due to worldly considerations like shyness invalidate the prayer? And what is the ruling on doing so, based on the statement of Qadi Iyad: "Leaving an action for the sake of people is showing off (riya')"?

Not vocalizing the recitation in prayer does not invalidate it, even if the reason is shyness, because vocalization is a Sunnah or recommended. A Muslim must not abandon what is commanded by the Shari'ah out of ostentation, nor should they act for Allah for the sake of people. And whoever is afflicted with shyness, they must treat themselves and get rid of it by remembering that Allah is watching during prayer. The aforementioned statement is by Fudayl ibn Iyad, not Qadi Iyad.