MC Martin Brundle suggested Hamilton wanted to give himself a few minutes to calm down before addressing the vast crowd, but the five-times British Grand Prix victor says it was because he had fought so hard to recover from last place he could barely stand.
That marked the second time in three races that a Ferrari had taken out a Mercedes on the first lap of a race.
"The mind-set we have tried to create in the team is that failure is allowed", Wolff, 46, added.
Their fierce rivalry on the track spilled over into a war of words in the paddock as Lewis Hamilton railed against what he called "interesting tactics" after a Ferrari collided with a Mercedes for the second time in three races at Sunday's British Grand Prix.
In France, Mercedes' management and Hamilton were among those to question why Vettel had only received the minimum penalty available to the stewards for causing a collision when Bottas sustained a puncture and dropped to the back of the field.
Mercedes team principal Toto Wolff said after the race that he was "not comfortable at all with the incident because it's tiresome being taken out in the first lap".
Arrivabene told Sky Italia after the race that Allison should be ashamed if he had said that.
"Today, he's doing his job".
The win was Vettel's fourth of the season, 51st of his career, and once-dominant champions Mercedes' first defeat at Silverstone since 2012 before the V6 turbo era started. "I locked the inside wheel a bit, couldn't slow down as much as I wanted, then we obviously touched and he spun".
The race had 15 laps left to run when it resumed on lap 37, but Romain Grosjean and Carlos Sainz crashed into each other one lap later, triggering a second safety vehicle that absorbed another precious three laps for the Mercedes drivers. Arrivabene suggested Allison should not blame others for his team's disappointment in front of a raucous 140,000 crowd.
"It's a tough day, to leave having lost points having been on the pole, but it is the way it is". Kimi had a good start.
Hamilton was sent spinning round, and suddenly he found himself down in 18th and with it all to do. Especially here because I mean, here is not easy. "I don't think there was an intention and l think it's unnecessary to even go there".
With Hamilton left with everything to do, it was Vettel that took the early march on the Mercedes of Valterri Bottas.
"Thank you for loving me for being with all my flaws", the 33-year-old wrote on Instagram. "And in a flawless world, I would have had new tires - but it wasn't that kind of day for me", Hamilton said.
"Accepting who you are and loving yourself are so important".
He said: "If somebody comes along and wants to pay an attractive price for anything, you have to put your commercial hat on and think about it commercially, not emotionally".
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