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.
Of course, they can see what Ten Hag managed to achieve at Ajax, what Manchester United was like with him, and what United became after he left. Obviously, managing such a crisis club is no easy task, but he handled it quite decently, even winning a few trophies. Bayer is a good team, and I think Alonso did a solid job there, so after him there won’t be a need to start everything from scratch, just to refine a few things. So yes, it’s quite possible that at Bayer, Ten Hag could achieve even better results than he did at Manchester United.