Ten Hag must fight to win over the fans, and that can only be done by leading Leverkusen to consistent victories. Meanwhile, regarding Ten Hag appointment as the new manager, Leverkusen management must have conducted a thorough evaluation of his capabilities. They wouldn't carelessly appoint a new manager, they clearly choose a manager who shares their vision. However, Ten Hag poor reputation with Manchester United is always the main narrative, maybe in Leverkusen, Ten Hag performance will be better.
Any new manager especially after tough time like his at Manchester United, has to win games mostly to get fans on his side. That is best way to change how people see him.
You are also right about Leverkusen management. Big clubs do not just pick new manager without thinking it through. They would have done their research looking for someone whose ideas match theirs and who they think can keep their recent success going. Even though people will still remember Ten Hag time at Manchester United new team and different goals and maybe more support could really help him to do better. Fresh start mostly brings out best in manager.
I think Ten Hag will be successful in this German adventure. A manager who won trophies with Manchester United will also be successful in Germany. The Bundesliga is a major goal for Leverkusen, and I hope they win. Bayern's wins every year make the league less exciting.