So your citation certainly does not prove DonQ's theory false.
Sitting shiva is first specifically mentioned in the Mishnah.
The Mishnah was composed of rulings of Jewish sages, many of whom were Jesus' contemporaries, so it makes sense that it was already an old practice at the time of the crucifixion.
According to Jewish law, only parents, spouses, children and brothers and sisters of the immediate family are required to sit shiva.
The practice is considered biblical and is traced to the seven-day mourning period observed by Joseph over his father in Genesis 50:10.
So, far from being "baloney", DonQ's thesis is certainly plausible.