Thomas, in fact, lost time for a second day running.
After saying he had overheated when losing time on Friday, he dropped another 29 seconds in the overall after being distanced by the lead group at the death.
"I just didn't feel quite on it from the start really to be honest - just quite weak," said the 33-year-old Ineos man, claiming he had limited his losses. "So I decided to go at my own pace."
For British outfit Ineos, however, their co-captain Egan Bernal can go into the most mountainous culmination of a Tour in years in full confidence after he finished just a few seconds behind the winner at the first of seven summit finishes over 2000 metres.
Pinot had vowed to attack this stage after losing time to an Ineos-led attack in a crosswind on the narrow road to Albi on Monday had angered him and his team.
Thomas has two Frenchman to worry about and is now two minutes and two seconds behind Alaphilippe in the general classification after he started to suffer with 500 metres to go on Saturday.
Alaphilippe was as surprised as anyone at Thomas wilting.
"I saw great riders blow up before me and in the end I went for the win, it's incredible," said the 27-year-old soldier.
French president Emmanuel Macron followed the whole stage from the lead car in the vast caravan and appeared animated as he was flanked on the podium by his two compatriots.
"This man has heart," he said looking at France's current national hero Alaphilippe.
On both the leader and Pinot, he added: "France is proud of them both."
As the 117km stage 14 climbed above the tree-line and above the mist where the oxygen levels were lower, contenders for the overall victory gradually fell by the wayside.
By the end there were just five men remaining with Dutch Jumbo Visma captain Steven Kruijswijk third at 6sec, with German Emanuel Buchmann and Bernal another two seconds behind.
Not only Ineos, but their rivals too may now switch their attention to young climber Bernal in light of Thomas's underwhelming show.
"Yesterday was one of the worst days of my career," said the 22-year-old Bernal whose slender frame is unsuited to time-trialling.
"Today I feel much better, the ascent was very hard and the best thing about it was managing to be amongst the best riders at the front."
Few expected the yellow jersey wearer Alaphilippe to thrive on Friday's time-trial, which he won, and fewer still expected him to survive on the Tourmalet.
But Alaphilippe dug deep and when he climbed out of the saddle to win the sprint for second, cheers for him were even louder than those for the popular Pinot.
But after 10 stages in the lead, the attention switched from the overall leader to the stage winner.
"Since the start of the Tour I've been targeting this stage, this is the one I wanted, the mythical Tourmalet," said Pinot, who is placed sixth overall, 3:12off Alaphilippe.
Pinot has long been thought a potential champion this year with the home nation waiting for its first Tour de France win since 1985.
Fans had been weaving up the Tourmalet since dawn and there was a carnival atmosphere with almost as many Basque flags as French ones.
Pinot, helped up the hill by young French champion David Gaudu, never faltered on the long final climb and his team boss at FDJ Marc Madiot was ecstatic at the finish.
Madiot had been upset by comments from Ineos boss Dave Brailsford on Monday when Pinot dropped down the rankings after getting caught in a crosswind.
"It feels good to twist the knife," Brailsford said on Monday.
Madiot was seen screaming "yes, yes, yes" at the finish line before racing off to embrace his team leader.
Leading Tour de France stage 14 results from Tarbes to Col du Tourmalet on Saturday:
1. Thibaut Pinot (FRA/FDJ) les 111.1 km in 3 h 10:20, 2.
Julian Alaphilippe (FRA/DEC) at 6, 3. Steven Kruijswijk (NED/JUM) 6, 4.
Emanuel Buchmann (GER/BOR) 8, 5. Egan Bernal (COL/INE) 8, 6. Mikel Landa
(ESP/MOV) 14, 7. Rigoberto Uran (COL/EF1) 30, 8. Geraint Thomas
(GBR/INE) 36, 9. Warren Barguil (FRA/ARK) 38, 10. Jakob Fuglsang
(DEN/AST) 53, 11. George Bennett (NZL/JUM) 58, 12. Alejandro Valverde
(ESP/MOV) 58, 13. Laurens De Plus (BEL/JUM) 1:19, 14. Richie Porte
(AUS/TRE) 2:05, 15. David Gaudu (FRA/FDJ) 2:26, 16. Enric Mas (ESP/DEC)
2:54, 17. Nairo Quintana (COL/MOV) 3:24, 18. Bauke Mollema (NED/TRE)
3:29, 19. Fabio Aru (ITA/UAE) 3:33, 20. Roman Kreuziger (CZE/DDT) 3:48.