They were wrong.
Instead, it was the start of a revolving door of kickers that hasn't stopped spinning. Since Gould, the Bears have employed five kickers over the past four seasons -- from Barth to Eddy Pineiro, with Mike Nugent, Cairo Santos and Cody Parkey in between.
All the while, Gould, 37, continued on with the San Francisco 49ers.
It's what can happen when a team moves on from a longtime successful kicker.
That had to play a part in what the Green Bay Packers did over the weekend, when they signed Mason Crosby to a three-year, $12.9 million contract less than a month before he would have become an unrestricted free agent.
If Crosby, 35, was tempted to try to finish his career with a warm-weather or dome team like former Packers kicker Ryan Longwell did with the Minnesota Vikings more than a decade ago -- and indications were Crosby was at least intrigued by that idea -- then the Packers wisely did enough to make that thought disappear.
In fact, it was just one day earlier when Packers general manager Brian Gutekunst admitted he would even consider using the franchise tag to keep Crosby around.
"I think when you have a guy who has been through the fire like Mason has, for a guy in my position it makes us feel very comfortable," Gutekunst said last week. "He obviously had an excellent year last year. He's a big part of our team, a big part of what we're trying to do here."
Any concern over whether Crosby's age could bring on a rapid decline didn't seem to scare Gutekunst as much as the prospect of going through what, say, the Bears have experienced.
At some point, Gutekunst will have to find the next Packers kicker. It took his former boss, Ted Thompson, two tries to replace Longwell. He first went the street free agent route with Dave Rayner, who lasted one season (2006) in Green Bay, and then drafted Crosby in the sixth round the next offseason.
Since the start of the 2007 season, no one other than Crosby has place-kicked for the Packers. In that same span, only two teams have used fewer than five kickers: Tennessee (four) and Indianapolis (three). Twenty teams have used more than five kickers, led by the Buccaneers (12) Chargers (10) and Saints (10).
Crosby has rewarded the Packers' patience on more than one occasion. They stuck with him through his slump in 2012. He took a pay cut the following offseason and then won a kicking competition in training camp -- the first he had gone through since his head-to-head with Rayner as a rookie -- to keep the job. He made all of his money back through incentives the following season and then signed a four-year, $16.1 million contract in early March 2016. Last year, that deal averaged out as the eighth-highest paid kicker in the league. His new deal, at $4.3 million per year, will make him No. 3 behind Baltimore's Justin Tucker ($5 million) and Gould ($4.75 million).
"I've been so fortunate to have a few contracts here; I've never hit free agency," Crosby said on Jan. 20, the day after the Packers' NFC Championship Game loss.
Perhaps Gutekunst didn't need to be reminded of the problems other teams have experienced when they have changed kickers as much as he needed to see how Crosby performed last season during difficult circumstances off the field. (His wife underwent surgery to remove a cancerous lung tumor in August and his brother's wife died of ovarian cancer in November.) Crosby went through another summer competition, beating out Sam Ficken, who ended up as the Jets' kicker. Crosby managed his highest percentage (91.7) on field goals in his career.
And if even Crosby loses some of his leg strength soon, his short-to-mid-range accuracy could still remain high. He's connected on 86.1% of kicks from inside of 50 yards in his career. Coach Matt LaFleur asked Crosby to try only five from 50-plus, and he made three of them. The analytics-driven coach might be more inclined to go for it on fourth downs rather than try long field goals regardless of who kicks for him.