It was not a matter of choice in the first place. Anakin was actually seen as a spy of opportunity; since Palpatine, for his own reasons, had started spending increasingly time with him, showed that he trusted him, and had just appointed him as his own personal representative on the Jedi Council - something that had never happened until then, since the Council elected their own members:
Palpatine: Anakin, I'm appointing you to be my personal representative on the Jedi Council.
Anakin: Me? A Master? I'm overwhelmed, sir
The Council accepted Anakin's appointment, but without extending to him the title of Master, something that Anakin, in turn, finds insulting, and helps to his gradual alienation from the Council and the Jedi Order.
So, since the Council had already a Jedi close to Palpatine, they decided to try to use it for their own benefit to the extent possible, by asking him to inform them on the Chancellor's dealings. Interestingly enough, they do not ask him in a formal session, but through Kenobi, in what would seem to be a private and informal conversation, had Kenobi not made clear that this was a Council request:
Obi-Wan: The only reason the Council has approved your appointment is because the Chancellor trusts you
Obi-Wan: Anakin, I'm on your side. I didn't want to put you in this situation.
Anakin: What situation?
Obi-Wan: The Council wants you to report on all the Chancellor's dealings. They want to know what he's up to.
Anakin: They want me to spy on the Chancellor? But that's treason.
Obi-Wan: We are at war, Anakin.
Anakin: Why didn't the Council give me this assignment when we were in session?
Obi-Wan: This assignment is not to be on record.
Anakin: You're asking me to do something against the Jedi code, against the Republic, against a mentor and a friend - that's what is out of place here. Why are you asking this of me?
Obi-Wan: The Council is asking you.
(To be honest, we never see the Council discuss this, and there is at least a hint that this may be a request not of the Council but of some inner circle - Mace, Obi-Wan, and Yoda.)
So, given this situation, there was never any option of Kenobi or anyone else actually stepping in and assisting or supervising Anakin in any way; it was a matter of Anakin being their only option for the task, no matter if some could not actually trust him for this. There was never any choice "we need to put someone close to the Chancellor - who would be appropriate?"; the initiative had always been with Palpatine here, starting with Anakin's peculiar and unprecedented appointment. And judging from the rest of the story, he played his cards too well.