Only in two occasions, in 2020 and 2014, have they failed to make a top-three finish in Paris, winning the race seven times between 2012 and 2019 through Bradley Wiggins, Chris Froome, Geraint Thomas and Egan Bernal.
Their decline, however, started in 2020 when then defending champion Bernal pulled out with back pains as Tadej Pogacar (24) emerged as the new monster of road racing.
Thomas and Richard Carapaz (30) finished third in 2022 and 2021 respectively but this year, the odds are long.
Rod Ellingworth, the team's racing director, however believes a podium finish on the Champs Elysees on July 23 is a reasonable ambition.
"We've got great capacity within the group and I think they're capable of making the podium if we get it right," Ellingworth told Reuters.
"I don't see why we couldn't. But we're looking at this race on two fronts. We want to take opportunities to try and win stages but also there's two GC (general classification) opportunities. One is Egan himself, we can't forget that his return to this race is phenomenal."
Bernal is making his comeback on the Tour after a career-threatening crash during training in Colombia last year and while no time frame has been set for him to get back to the very top, he knows where he wants to go.
"We should praise the lad for what he's done. I'm really proud of the team and the support staff for what they've done but it's all about him, what he's done to get back. He wants to win this race again," said Ellingworth.
Bernal took a decent 12th place overall in the Criterium du Dauphine last month and finished Saturday's first stage 33 seconds off the pace, not too far behind the biggest race favourites.
"He's got his feet firmly on the floor now but I tell you he wants to win this race again and we want to go on that journey with him. That's really exciting for us," Ellingworth explained.
"In terms of numbers he's not where he was when he was winning the race but he's not a million miles off. What is a bit different now is that the margins are really small these days but we know with Egan, he's a professional, he's a great leader and I think he'll do pretty well here."
The other option for Ineos-Grenadiers is the maverick Pidcock, who claimed a spectacular win at l'Alpe d'Huez on his Tour debut last year.
"Now he wants to take the next step and the next step is consistency and we shall see. It's an opportunity (for the team) to move forward," said Ellingworth.
"We're not coming in with a strong GC (leader) and that's unusual for us but life takes its route and sometimes you can do all the planning in the world and it doesn't quite happen. So we're quite excited about the race, in a different way."