Vettel and Ferrari fought for the 2018 world championship but were defeated again by Lewis Hamilton and Mercedes after a string of high-profile errors.
Speaking at the 2019 Autosport International Show, Brundle said he did not like to get “personal” with his assessment because “you don’t know what you don’t know if you’re not inside an organisation”.
“But I did observe it, and see that things weren’t right,” Brundle said.
“You look at situations like Hockenheim, where Vettel was put under pressure because they didn’t do the right things earlier on in the race.
“Then he fell off the road, and he fell out of the championship from that moment onwards.
“You look at other things like the slipstreaming in Monza [when Vettel did not get a tow off Kimi Raikkonen because Ferrari stuck to a previously-agreed order on-track].
“Certain things weren’t being done.”
Arrivabene has been criticised for how he handled the media in the latter half of 2018, particularly after the death of Ferrari president and CEO Sergio Marchionne, who preferred to be the team’s spokesperson.
Brundle said the highly-rated Binotto, who will work under John Elkann (president) and Louis Camilleri (CEO), would need to adapt to such challenges in his new role.
“I do think Mattia Binotto has got a very good reputation in developing the team technically, he’s clearly a leader of people,” said Brundle.
“But of course now he’s the lightning conductor. He’s got to be up front, he’s got to speak to nasty, horrible people like me in the Formula 1 media.
“He’s got to be there and explain why things went well, why things didn’t go so well. It’s a different job and a different challenge.”
Highly-rated rising star Charles Leclerc has joined Ferrari in place of Kimi Raikkonen, after just one season in F1 with Sauber.
He has been tipped to win races in his first year at the Maranello team and put four-time world champion Vettel under pressure.
If Leclerc can join Vettel in fighting Mercedes and Hamilton from the beginning, Brundle believes Binotto will make life simpler for Ferrari.
“I do know from people like Ross Brawn and others who have worked with him, he is most highly regarded,” said Brundle.
“I think he’ll get the car and the engine to where it needs to be and the rest of it should be easy, if Vettel and Leclerc are doing their jobs.”