The account of the image's arrival is a pious tradition, reported as such by Vatican News; the Church has not defined it as historical fact, and devotion to Mary under this title does not depend on it.
What is documented is a long papal connection: Urban VIII entrusted plague-stricken Italy to her in 1630, Bl. Pius IX made a pilgrimage in 1864, St. John XXIII came in 1959 and St. John Paul II in 1993, before his journey to Albania. Leo XIII raised the church to the rank of minor basilica in 1903 and added "Mother of Good Counsel" to the Litany of Loreto.
Genazzano lies outside Rome, so a visit needs its own transport and time set aside. Check with the shrine directly for opening and liturgical times before travelling. It is a Catholic pilgrimage place, not a current Bold Pilgrim itinerary stop: no dated departures are currently announced.