Why can't Anakin be a Master?
Anakin's appointment was political, not spiritual. Historically the Jedi elected their own leaders by consulting the will of the Force and by mutual consent. Anakin, on the other hand, was appointed directly by Palpatine as his 'envoy' to the Council, and his appointment pushed through by Palpatine's allies in the Senate.
In short, the Jedi don't think he's ready. He's far too young, hot-headed and well, arrogant, to be part of their elite.
“Then we should use someone else’s eyes,” Obi-Wan said “Forgive me,
Master Yoda, but you just don’t know him the way I do. None of you
does. He is fiercely loyal, and there is not a gram of deception in
him. You’ve all seen it; it’s one of the arguments that some of you,
here in this room, have used against elevating him to Master: he lacks
true Jedi reserve, that’s what you’ve said. And by that we all mean
that he wears his emotions like a HoloNet banner.
Revenge of the Sith: Official Novelisation
On top of all of that, they really don't trust him. They think his loyalties are divided between the Jedi and the Chancellor.
“Listen to me, Anakin,” Obi-Wan said, and the seriousness of his tone caught Anakin’s attention. “The fact is, you’re too close to the Chancellor, and the Council doesn’t like him interfering in Jedi affairs.”
Revenge of the Sith: Junior Novelisation
There's also the fact that the rank of Master gives you access to certain artefacts such as holocrons and the deeper teachings of the Jedi. In the wrong hands, those would be spectacularly dangerous.
All that mattered was Mastery.
All that mattered was Padme.
This was a gift beyond gifts: as a Master, he could access those forbidden holocrons in the restricted vault. He could find a way to save her from his dream …
Revenge of the Sith: Official Novelisation
So why did they agree to have him on the Council at all?
His appointment was a done deal. They weren't consulted on this, they were simply informed that Anakin had been made a Council member. It's not a hill that they're willing to die on, so they have to wear it. On top of that, having him on the Council means that Palpatine will consult with him more than ever before, which gives the Jedi an opportunity to have Anakin spy on the Chancellor for them and to feed him misinformation.
Obi-Wan stopped walking and turned to face Anakin. He hesitated, as if
he was searching for the right words. Then he took a deep breath. “The
Council wants you to report on all of the Chancellor’s dealings. They
want to know what he’s up to.”
Anakin stared at Obi-Wan, stunned. A tiny part of his mind whispered,
Don’t ever ask Obi-Wan to come right out and say something, ever
again, but most of him was trying to absorb what Obi-Wan had just
said. “They want me to spy on the Chancellor?”
Obi-Wan nodded.
Revenge of the Sith: Junior Novelisation