Some powers within the Bleach series are simply just too dangerous, too powerful or too unpredictable to be used in the presence of others. Sosuke Aizen barred Ulquiorra Cifer from releasing his Resurrección, Murciélago, under the ceiling of Los Noches, while Shunsui Kyoraku is discouraged from using his Bankai, Katen Kyokotsu: Karamatsu Shinju, due to the volatile nature of its power. However, Shinji Hirako's Bankai is banned in the entirety of the Soul Society due to the immense danger it poses to any caught within its indiscriminate sphere.
In the Can’t Fear Your Own World light novels, written by Ryogo Narita and co-written by Tite Kubo, a few details that were left unanswered in the Bleach series were explored -- one of these being Shinji’s Bankai. During the Gotei 13’s confrontation with the Tokinada Tsunayashiro, the novel's antagonist, Shinji is given the perfect opportunity to unleash his Bankai. Because of their location and the situation Shinji finds himself in, he isn't allowed to do so. Even though the Bankai is described in a written format, there is an image in the light novel used to portray its appearance, as well as the mobile game, Bleach: Brave Souls, which has a usable Shinji character with his Bankai power.
How Shinji's Bankai Works
Shinji’s Bankai, Sakashima Yokoshima Happofusagari, transforms his appearance into a staff with the same circular pommel as his Shikai and causes large flower petals to erupt around and then encase him. This protects Shinji from the effects of his Bankai, which is unforgiving in practice. Anyone within the vicinity of his Bankai is affected by its power, turning friends against friends and foes against foes, forcing people to kill each other without mercy. All are affected without exception, making Shinji's Bankai one of the most effective when fighting multiple enemies. In the light novel, Shinji utilizes this power to wipe out an endless horde of hollows, showing how useful it is against streams of enemies but its lack of usefulness when confronting single foes.
Because of this particular power, Shinji's zanpakuto is useless in one-on-one fights. This makes the lack of its usage during his fight with Aizen all the more reasonable. While it may have been useful if Shinji was facing Aizen and his entourage by himself, the fact that Ichigo Kurosaki, the Gotei 13 and the Visored were all present alongside him in Fake Karakura Town meant that Shinji was unable to unleash the full might of his power. Even if Shinji had wished to make use of his Bankai against Aizen, Sakanade would have refused him -- Sakanade will only release into Bankai form when Shinji is facing two or more enemies due to Sakanade’s own preference in combat.
Shinji's Bankai Could Appear In Thousand-Year Blood War
It is important to point out that Shinji’s Bankai has only made an appearance in the light novels, which is debatably canon. Kubo did have a hand in the creation of the novels as a credited co-writer, which lends to the inclusion of Shinji’s Bankai being canon, and it was perhaps included due to Kubo’s lack of time to unveil everything he wished to in the manga. Now that the anime has continued with Thousand-Year Blood War and the details of Shinji’s Bankai are known, it's possible that the anime will adapt a battle for Shinji in the second part and give him the limelight to brandish his terrifying Bankai.
With all these factors taken into account, it makes sense that Shinji’s Bankai has been barred within the Soul Society. The risk of it causing harm to his fellow Soul Reapers is far too great, and it shows just how powerful Shinji truly is. The reappointed Captain of the 5th Division has been regarded as fairly weak in the past comparatively with some of the more popular captains in the series, but should he be unrestricted with his Bankai, he could easily turn the entirety of the Soul Society into a bloodbath.