Sutton cited Giggs’ Premier League experience – the 42-year-old made 963 appearances for United as a player and was assistant manager at Old Trafford until July.
Bruce, 55, has been linked with the England job since leaving Hull, the club he led to two promotions and an FA Cup final, in the summer. He is also being linked with Aston Villa.
Swansea, who were taken over by Americans Steve Kaplan and Jason Levien in July, interviewed former United caretaker boss Giggs for the job.
Ex-Egypt and US coach Bradley is joining from French second division side Le Havre, and stayed to oversee their 2-1 win over Sochaux on Monday.
Swansea have lost their past three league matches and are above the relegation zone only on goal difference.
And while ex-Chelsea striker Sutton does feel Bradley lacks the Premier League experience needed to guide them away from trouble, he agrees they were right to let ex-Udinese and Monaco boss Guidolin go just seven games into the season.
“Swansea for me, since Guidolin has taken charge, have lost their identity and their way of playing,” he told BBC Radio 5 live.
“They’ve been bold and made the decision early and I have no problem with that.”
Meanwhile, former Swansea winger Leighton James believes the club decided to replace Guidolin before Saturday’s defeat by Liverpool.
It was Swansea’s sixth Premier League game without a win but James believes the club’s board had already decided to act.
“This was a done deal before Saturday,” James told BBC Radio Wales. “There’s no doubt in my mind Bob Bradley was coming in this week.
“But what would have happened if we’d have won on Saturday against Liverpool as we looked like doing at half-time?
“We’ve got so many games left to go. I personally think it’s the wrong time.”