The Portuguese took the reins in May as the replacement for Sam Allardyce, who was sacked shortly after the end of the 2017-18 season.
Silva had been linked with a potential move to Everton a few months earlier following the sacking of Ronald Koeman in October, but the job eventually went to Allardyce as Watford refused to let the former Hull City boss speak to the Goodison Park club.
"I don't understand Everton's obsession with Silva and I hope he proves me wrong," Cottee told the Liverpool Echo. "He was relegated with Hull and was sacked after a poor run of results with Watford.
"Other managers could have come in and done as good a job, so I think he needs to prove himself still. He's got a lot to prove to the fans. They will be looking for something different after what happened to Sam Allardyce as well with Koeman before him.
"They need a period of stability and I hope Silva does that, but overall, I don't see the big obsession with his appointment.
"He hasn't won any major honours in the big leagues around Europe, so Everton has got someone with potential, rather than the finished product."
Everton endured a tricky season last time out, initially struggling but ultimately finishing eighth in the Premier League table amid heavy fan criticism of Allardyce and the club board.