Adding to this, as I agree with OP's commentary:
Anakin became a member of the Jedi High Council because Sheev Palpatine pressured the High Council to make Anakin a member of this body. This was, accordingly, a highly-politicized internal promotion of a member of the Jedi Order.
Sheev Palpatine has never been a member of the Jedi Order, and had no right to force the High Council to internally promote someone based on his wishes. It's debateable if the Republic, the Senate, or the Chancellor even had the legal authority to make such a demand.
This is the reason why there was so much tension about Anakin's placement on the Jedi High Council.
It is perhaps only a mark of self-preservation of the Jedi Order itself that Anakin was not also shunted into the rank of Master by means of an outsider's political influence.
A Jedi Knight is granted the opportunity to be a Jedi Master upon sufficient proof of what is effectively self-mastery - demonstrated typically by means of one's masterful understanding of the Force (they're Jedi. Connection to the Force is the basis of their religion.).
Knights typically prove their mastery of the Force, historically in-universe speaking, by raising several Padawans successfully into knighthood themselves. This means the Padawan in question must complete what are typically four traditional trials:
BreezeWiki link further detailing the trials mentioned above here.
What this would mean in the context of Anakin Skywalker's opportunity to be granted the rank of Master, firstly, is that Ahsoka Tano would need to pass this trials. At a minimum. Typically one would need to raise several Padawans that also pass these trials.
Something that would ideally be obvious for any reader of this post that is also a fan of Anakin, that also believes Anakin should have been given the rank of Master because he was pushed by way of Chancellor Palpatine's political manipulations of the Jedi High Council to obtain a seat on the High Council, is that Anakin himself needs to ALSO pass these trials.
Here is a description - all of these will be from BreezeWiki - of the Trial of Skill:
The Jedi Order required its members to show a basic degree of competency with the lightsaber and an ability to use the Force to affect and enhance themselves for the purposes of combat. Jedi Master Yoda subjected Luke Skywalker to a Trial of Skill on Dagobah.[8]
Does Anakin pass this trial? Yes ✅
Here is the Trial of Courage:
The trial involved the Jedi confronting their greatest fears. The apprentice must strive for fearlessness, even in the face of certain death. Such fears usually involve death or a personal failure that leads to the death of someone else.[8]
Does Anakin pass this trial? NO ❌
Twice over! He committed a genocide of the Tuskens on Tatooine because of his mother and her subsequent death, and then he committed a genocide of the Jedi because of his fear of Padmé's death in childbirth! Anakin does not pass this trial!
Here is the Trial of the Flesh:
The trial historically involved physical pain, but was later adjusted to focus on sacrifice and commitment to the light side. The trial involved the apprentice being forced to choose between protecting the objects of their attachment or sacrificing themselves and the beings they are attached to for the greater good.[8]
Does Anakin pass this trial? NO ❌
How many people does Anakin let die in order to maintain his attachments, or else directly commit murder in order to maintain his attachments? He already commits genocide twice over in canon! This is not even going into issues he has during the war itself, such as prioritizing R2D2 over the safety of his troops, or even failing to come to Obi-Wan's aid in a timely manner when Obi-Wan discovered the clone army in the first place on Kamino, thus directly leading to Obi-Wan's capture in Geonosis and likely also torture before he was slated for execution.
Here is the Trial of the Spirit:
"Your mother was taken from her planet as a child as soon as she showed signs of Force sensitivity. She was raised in the crèche, just like you, and grew to be a middling Padawan. She did not complete the Jedi trials. It was the Trial of the Spirit that got her. She lacked focus and concentration, couldn't control her mind. Just like you, Iskat. But we'd been too easy with her, had assumed her species simply matured more slowly. That was our mistake. She left questioning everything, unsure of herself. It was as if her spirit had broken. Although she was offered civilian work in the Jedi Temple, she chose to walk away entirely. She was escorted back to her homeworld and delivered safely to her family."―Klefan Opus, to Iskat Akaris[9]
The trial pits the apprentice against their own shortcomings and was one of the best trials to conduct using a vergence. It was designed to give the apprentice an opportunity to resist their emotional weaknesses, giving them a firm moral footing from which to resist the darker aspects of their characters.[8]
The Padawan Feyra Akaris left the Jedi Order after failing her Jedi Trials, specifically her Trial of the Spirit.[9]
NB: Here is a definition of a vergence, as cited by BreezWiki (link here):
A vergence,[2] sometimes described as a Force nexus,[3] a nexus,[4] or a locus,[5][6] was an unusual yet naturally occurring concentration of Force energy localized around a place, object, or person. A vergence centered on a location could give nearby Force-sensitives stronger interactions with the Force, often including visions.[7] The ethereal world of Mortis was the fulcrum of the galaxy and the Force.[8]
Does Anakin pass this trial? NO ❌
This is ably demonstrated by Anakin's Fall, which resulted in him leading the march on the Jedi Temple to genocide the Jedi. A follow-up action to his Fall would be his weakness in refusing to assist Mace Windu in felling Sheev Palpatine, due entirely to his fear of Padmé dying, and his agreement to killing an disarmed opponent (Count Dooku) merely upon Sheev Palpatine's insistence.
Could Anakin have refused to commit genocide? Yes. Did he? No.
Could Anakin have refused to kill Count Dooku? Yes. Did he? No.
Here Is the Trial of Insight:
The trial tested the apprentice's ability to resist illusions and deceptions. It also often included some kind of riddle or puzzle for the apprentice to solve, with an answer that was further clouded by illusions.[8]
Does Anakin pass this trial? NO ❌
Quite the fuck obviously not, he does not pass this trial. He did not once question either his vision of Shmi's death or of Padmé's death. He simply assumed these were iron-clad depictions of the future, that were invariable, unchangeable, and immutable, and made decisions based entirely on these untested assumptions.
Does he even ask for help or seek guidance when he sees something disturbing, that he presumes is a vision? No! He does not! He assumes that he is correct, and that his interpretation is the only correct interpretation to be had.
He doesn't even go pick up a book about it! He had a whole Jedi Temple full of resources on the Force - about many aspects of the Force, even! - and he not once goes to look things up.
Anakin Skywalker did not so much as once in canon prove his capability in handling the responsibility of being granted the rank of Jedi Master. Ergo, the Jedi High Council was entirely correct in refusing to name Anakin Skywalker a Jedi Master.
