Fitzpatrick is on thin ice as it is, just knocking off Murphy and retaking a seat that used to be his. People are lukewarm to him, they just hated Murphy way more.
They had groups of supporters, and wanted the supporters to be with them when they were “sworn in”. So they did it while watching on the TV, so their supporters could be there.
They already had signed the official oath, a written form collected by the clerk of the house. I’m pretty sure the written document is the official document that shows they took the oath, and the public oath is theater. They just pushed the theater to a new level.