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.
You're right, Leverkusen management did not judge Ten Hag because of his poor performance in the Premier League, but they also do their research to see if the manager has a history of winning titles, Ten Hags won two trophies in two seasons at Manchester United, and we all agree was an achievement because look at Arsenal, no manager has won any of the trophies that Ten Hags has in recent seasons. If we look at his former club's record before arriving at Manchester United, he is an excellent manager for the team.
However, the Bundesliga is different from the other leagues, and the manager can find the Premier League difficult. The Bundesliga is not as competitive as the Premier League; perhaps we will see different results from Ten Hag, but I'm not just impressed with their current players, we may not see them pressure their opponents like they did under Xabi Alonso.