Team Sky, another unusual early starter, were booed by roadside spectators throughout their effort but posted nearly identical times through the first two time checks - as did BMC Racing, who also found themselves in an unfamiliar early start time.
"We expected some crashes in this tricky final and moved up early", said Enrico Poitschke, sports director of Sagan's Bora-Hansgrohe team.
Named the "most sporting city of France" on three occasions (1972, 2007 and 2014) by sports newspaper L'Equipe, Cholet has hosted every type of Tour stage: a team time trial in 1936, a normal road stage in 1998 and an individual time trial in 2008.
Froome was left 55 seconds behind in the overall standings with another week of nervy rolling stages before hitting the Alps.
While Sky's top placed rider is Geraint Thomas three seconds off the lead, 2017 Giro victor Dumoulin is just 11 seconds down, but perhaps Bardet is the man with the most to smile about at this stage.
"We're just. trying to get through the best we can until we get to the mountains", Froome said.
At 1.8 kilometres from the finish line, there was a crash that involved the Yellow Jersey wearer Fernando Gaviria (Quick-Step Floors) as well as Michael Matthews (Sunweb), Daryl Impey (Mitchelton-Scott).
Yates and Froome sit in 81st and 84th places respectively, 67 seconds off yellow but more concerned about the 51 second gap to the likes of Bardet (AG2R La Mondiale), Vincenzo Nibali (Bahrain-Merida) and Mikel Landa (Movistar). "We're on track but there's still so much to go through".
"It's amusing because at the end of the day it's just another bike race. You have to step up and deliver on the day and that's not an easy thing to do on a Tour de France team time trial stage".
The Tour heads into the cycling hotbed of Brittany on Tuesday for Stage 4, a mostly flat 121-mile leg from La Baule to Sarzeau that starts and ends on the Atlantic coast. "That's just the nature of the game".
"They were the second to last team and they had a great ride".
The 39-year-old, competing in a record 18th Tour de France and his 350th stage, enjoyed his day in the spotlight, sitting up to high-five fans in the towns early in the day as the peloton was happy to let him go.
"You probably won't believe me but I didn't even think about it", he said of yellow.
"But even if you're on the course and you are taking time back, it's so technical and you have to nail every corner and every turn", he said. We just wanted to get that stage. "Everyone is working well together and we have a strong team and Richie can be up there".
For Porte, the stage win took on special significance as it marked a year since he crashed out of last year's Tour on the decent into Chambery on the ninth stage. "I'm very happy with the other riders".