The 24-year-old has thrown his hat in the ring to deputise for skipper Mills after his midfield performance led the Swans to a stirring qualifying final win over Brisbane.
Sydney coach Dean Cox also has an experienced leader in former co-captain Dane Rampe, as well as fellow veteran Jake Lloyd, midfielder James Rowbottom and Tom Papley at his disposal.
Whoever Cox lands on could also play a key leadership role next year, with Mills facing the possibility of missing the entire 2027 season after tearing the anterior cruciate ligament in his right knee.
Swans vice-captain Isaac Heeney is among five players serving club-imposed bans after last month's post-match incident at the Pullman Hotel in Melbourne.
Star defender Mills had earned significant praise for leading the Swans through the turmoil, which past players said shattered the club's famed "Bloods culture".
Chad Warner, Nick Blakey, James Jordon and Riley Bice are the other Swans who won't play again this season.
Gulden rallied the Swans after Mills went down late in the first quarter against the back-to-back reigning premiers, finishing with one goal, 34 disposals and eight clearances.
"If it (the captaincy) was offered to me, I would love to," Gulden told AAP.
"I would feel like I'd be ready. I wouldn't change anything that I'm doing. I don't think any of us need to change anything that we're doing.
"But you're just so shattered for the person. He's (Mills) just an amazing leader.
"He is the best human being at the club. It's just not fair.
"We've just got to keep going, and it's going to be next man in."
Gulden said the 53-point qualifying final win left him with the most pride he's ever felt in the Swans across six seasons at the club.
The two-time All-Australian has played in two losing grand finals (2022, 2024), with last year's 10th-placed finish the first time he had missed finals in his career.
"I love playing with this team. The last couple of games have been so fun as a player to be a part of," Gulden said.
"There's something in having your back against the wall and trying to fight your way out of it.
"The weekend (against the Lions) was easily the most rewarding game.
"The most pride I've felt to be a part of this team in my career.
"There was obviously some respect lost for the club. It's completely fair enough.
"All we can focus on as players is what we can control, and that's going out there and representing the club the best way we can on and off the field.
"I'm really proud of the club and the team, everyone involved, because the last few weeks have been really difficult."
Sydney will continue their unlikely flag push by hosting minor premiers Fremantle at the SCG on Friday.
"A premiership would mean the absolute world to me," Gulden said.
"Even if what happened hadn't happened, it still would mean the world to me."
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