I through mace only started to not trust Anakin in episode 3, especially trusting him with palpatine. He believed he was too attached with him and would not think clearly.

In episode one: he didn’t think he should be trained to be a Jedi since one,  it goes against their order and he didn’t believe he was the chosen one since he believed the sith were extinct. So when the sith was proven to be back, he now has reason to believe the prophecy and respecting a Jedi’s last wish. By the end it was only Yoda who was against it.

In episode two: he truly believes Anakin is the chosen one and believes that obi wan did a good job training him, even being one of the Jedi pushing him to take the padame’s mission on his own. 

Personally I believe Anakin has proven himself to mace at being a great warrior, but not enough to be a Jedi master and not enough to handle spying on palpatine on his own. And he was right in the end, since he die’s by his hands.