Australia's two young lions Lucas Herbert and Cameron Davis both made the cut and comparative veteran Adam Scott made a run, but they are all chasing Americans Zach Johnson and Kevin Kisner at the Open Championship.
"Who hasn't?" Spieth said after seizing upon a calm Carnoustie for a 6-under 65 to tie for the lead with Kevin Kisner and Xander Schauffele.
Defending champion Jordan Spieth carded a 67 to move three under.
He added: "I feel like there are low rounds in me". Playing in the third group on the course he needed only 64 to set the Open Championship course record at Carnoustie.
After a frustrating first round, Spieth bounced back with a fine 67 on Friday, one of the best rounds of the whole day featuring only one bogey, at the 16th.
The seven occupants own eight major titles between them, with only Kisner and Fowler yet to get off the mark.
"I wasn't in a fraternity in college but it kinds of feels like I'm going back to my alma mater and I'm the old guy stepping into the current frat house", said Johnson, who also won the Masters in 2007. The greens held shots a little better.
Can Fleetwood or McIlroy end American dominance?
More telling than that succinct summation of a day that started wet, got wetter but ended dry came when Woods was asked to identify the one shot he would like to play over.
Matt Kuchar, who was denied a maiden major by Spieth at Birkdale previous year, moved to four under with a 68 alongside compatriot Tony Finau and South Africans Erik van Rooyen and Zander Lombard, who all shot level-par 71s. The leaders teed off at 4 pm ET Saturday afternoon as Troy Merritt leads Billy Horschel by two shots entering Round 3. Woods had another 71 and was six shots behind, with 28 players between him and the lead.
Fleetwood and McIlroy were playing a few groups ahead of Johnson, and it was a race to see who would wind up on top for much of the late morning until Johnson made his birdie putt on the final hole.
Carnoustie's finishing stretch also killed off the challenge of Justin Thomas, the man closest to him in those rankings.
The last world number one to miss the Open cut was England's Luke Donald at Royal St George's seven years ago.
Rory Mc Ilroy of Northern Ireland plays from the fairway in the second round of the British Open on Friday
Moments later, Woods began his charge to get into contention at a major for the first time in five years.
There was not a single bogey in Fleetwood's round, while the last of his six birdies came at the 18th.
"It was a struggle on Thursday", he said.
"I certainly am not the No. 1 player in the world that ends in 'Johnson", he said.
"It's good for golf".
Pep Guardiola has picked Rory McIlroy and Tommy Fleetwood as his favourites to win The Open this weekend along with a having a watchful eye on Rafael Cabrera-Bello. Do I think he's going to win on Sunday?
The calm, baked conditions that have swarmed Carnoustie this week were initially replaced by heavy downpours more associated with the Championship before reverting to the week's type later in the day.
Lucas Herbert rides a birdie putt into the hole during the third round.
"That's a great way to finish", he said.
"I managed to stay away from crying, but definitely had a lump in the throat". He didn't miss the cut so it will be good.
On a day ideal for low scoring on Saturday, Woods let rip with his driver and tucked into the birdie buffet, briefly grabbing a share of the lead - something the American had not done since the 2013 Masters.
Rory McIlroy is still pursuing that aggressive attitude and aggressive gameplan and he is looking to play to his strengths - he is not being cowed by the speedy nature of the fairways. "I love where my putter is and love my position going into the weekend".
'I'm happy for Spanish golfer Rafael Cabello.