If Bayern wanted to keep him they easily could have, especially when they are financially stronger than Liverpool. No way Bayern accept €30m for him if they don’t see him as surplus to requirements.
He would've left in 2021 for free, if they kept him this year.
This move wasn't about money etc., it simply was about a new challenge in a new league. Bayern and Thiago made that clear. It's also why they respected his wish and let him leave for around €30 million this year (sign of respect for Thiago and at least getting something for him).
As far as his reputation of "playing bad in big games": He certainly played really well in the CL final this year and he's probably the best technician in the Premier League.
This reputation isn't really coming out of nowhere though. Before this CL tournament, he really has been merely "solid" or at times even "invisible" in most big games over the years. There is a reason nobody really talked about him before those Liverpool rumors surfaced and he played lights-out in the CL tournament.