He was the son of Balthazar Francis Xavier, Councillor, bailiff and Diet envoy, and Anna Klara Achermann.
Wyrsch began his art studies 1745 as a portrait painter with Johann Michael Suter in Lucerne and Franz Anton Kraus in Einsiedeln as teachers.
After his art studies in Italy, he returned to Switzerland and began his artistic activity as a portrait and church painter.
With an increasing blindness ascribed to cataracts, he withdrew to Buochs, where at the Conquest Nidwalden he was murdered by the troops of Napoleon Bonaparte.
In 1780 the abbot of Disentis, Columban Sozzi, paid attention to the talent of Felix Maria Diogg (1762–1834), and enabled him to travel to Wyrsch in Besançon.