Former world No.1 Woods was in the running to claim a record-equalling fifth PGA Championship and his first major since the 2008 US Open, but fell short by two strokes.
Furyk said Wednesday he was open to the idea of Woods serving double duty as a player and vice captain but wanted to see what worked best for the team if that possibility arose. That one went in.
"I'm pretty disappointed with myself that I couldn't play the last three holes better."
The quirky and analytical DeChambeau has become a player Woods has taken to.
The two true wild cards after Woods and Mickelson are much more up for debate. But remember he was late to the party having only returned to regular PGA Tour action at the start of this year.
The main issue will be whether he retains his position as one of Furyk's right-hand men. "It's a fun atmosphere for us to play in".
"Other than me, my team, everybody was rooting for Tiger", he said.
"To do it in this fashion, playing against Tiger and Scotty down to the wire, it's really incredible".
And imagine the furore and potentially unsettling impact of leaving out such an iconic figure.
Woods has been in the mix a few times but his Sunday performance, particularly on the back nine, has not been up to his once unassailable standard.
"I don't think I'd ever played in front of that many people", he told reporters Saturday. "They can be the calm in the midst of a storm". They'll be hoping that he leaves his "A" game Stateside, however. "It would be hard". "I do want to be there as a player", Woods said Sunday.
Now, early next month after each of the FedExCup playoff events, Furyk will select his four captain's picks.
Here, we take a look at the names in the frame.
"We are looking at how our next four fit in with our eight qualifiers and how that team meshes and gels".
"I mean, everybody on the golf course heard it (the noise)", Koepka said.
"I'm not sure the numbers are always that important when I look down the list", Furyk said.
Woods has gone 3,709 days since his most recent major win with more to come until his next chance at the 2019 Masters next April, the 43rd major since his last win.
Koepka successfully defended his breakthrough 2017 US Open victory at Shinnecock Hills this year to win back-to-back US Open crowns.
Koepka is proving to be the prototype player for his generation and beyond - a 28-year-old workout nut who splits time between the weight room and the driving range.
"He seems to rise to the occasion", said Furyk.
It would be the second time he plays the Ryder Cup as a captain's pick.
Koepka also has experience of playing the French Open at the venue for this autumn's match.
The unorthodox 24-year-old then endured a remarkable collapse in the closing stages of the Porsche European Open, but with Justin Thomas now the only rookie in the USA team, Furyk would have no qualms about giving DeChambeau a debut, providing he can find some form over the first two events of the FedExCup Play-Offs.