Why didn't Mace Windu think Anakin could handle being near Palpatine?

Question

Mace Windu had doubts about whether putting Anakin could handle being near Chancellor Palpatine.

OBI-WAN: "Anakin did not take to his new assignment with much enthusiasm."

MACE: "It's very dangerous putting them together. I don't think the boy can handle it. I don't trust him."

_{Star Wars : The Revenge of the Sith}

Why would he doubt Anakin's abilities? Anakin was a fully trained Jedi by then. If Anakin was not ready, he should have assigned somebody else. He was aware that Anakin had singlehandedly rescued not just Palpatine, but also an unconscious Obi-Wan from General Grievious shortly before that, so he should have considered Anakin as fully capable of being a bodyguard. Mace had no idea at the time that Palpatine was a Sith Lord, so he should not have feared a Jedi being near him.

Did he foresee something? If he did, why not assign a different Jedi knight to guard the chancellor?

Answer 1

Windu is not just doubting Anakin, but Palpatine too.

1. Both Yoda and Windu and the majority of the Jedi Council were against of training Anakin from the very beginning. He was too old to be taken in as a youngling, which meant he was too emotionally connected to the family, that too very strongly because the only family he had was his mother. Anakin was worried about his mother all the time even after being taken in as a Padawan (which I don't disagree with). This is shown in multiple scenes, implying he has never been able to cut emotional ties. However, to be a perfect Jedi, you never should never allow your emotions take over the logic.

Fear leads to anger… anger leads to hate… hate leads to suffering

It was the death wish of Qui-Gonn and the level of confidence of Obi-Wan after beating a Sith(Darth Maul) that made the Jedi Council reluctantly let Anakin to be trained. It is plausible Windu never dropped his doubts since then.

2. Mace Windu was skeptical as a person. Watch his tone even after Anakin revealed his suspicions about Palpatine being a Sith Lord.

3. There's another factor of Anakin being seriously talented. His force potential(his Midi-chlorian count, to be exact) was higher than Yoda, even as a child. Which means if he ever screws up, the result is going to be catastrophic.

4. Palpatine was already under suspicion of the Jedi Council for some time. They might not have suspected he was The Sith Lord Darth Sidious they were looking for, but they obviously were critical of his recent actions and the level of omnipotence he has gained after becoming the Supreme Chancellor. So there's Palpatine who's related to shady business who's personally asking for Anakin, the hot-headed teenager who's unbelievably talented, you're obviously going to suspect that something is not right.

If the first 3 points are related to Windu doubting Anakin's capability as a Jedi and the 4th point is related to Jedi Council suspecting Palpatine.

If you carefully read this quote by Windu again

"It's very dangerous putting them together. I don't think the boy can handle it. I don't trust him."

It is very possible Windu is referring to Palpatine, by the word him. Therefore Windu is not just doubting Anakin. His main concern is Palpatine.

Answer 2

Throughout Anakin's time as a Jedi he and Palpatine had formed a close friendship, close enough that Anakin had told the Chancellor about the circumstances of his mother's death and his subsequent massacre of the Sand People responsible, while the Jedi wouldn't have been aware of the exact content of such a conversation the trust between them would have been noticed.

There's also the very reason they wanted someone to spy on Palpatine in the first place, even in dealings that weren't entirely buried by his schemes as Sidious the Chancellor had successfully manipulated thousands of different politicians to grant him ever greater power over the course of the Clone Wars showing that he might even be shrewd enough to manipulate a Jedi, particularly one as shall we say straightforward as Anakin.