The Band of Aura is only meant to help with the upkeep of early-game trinkets. Its mana production rate is extremely low compared to any other source of mana in Botania. It manages to supply a Sojourner's Sash (or its upgrades), but that's it.
The Greater Band of Aura produces mana at 5x the rate of mana the basic Band of Aura, and a complete terrasteel armor set produces mana at double the rate of a Greater Band of Aura. Wearing the armor and one Greater Band of Aura gives you about the mana production rate of a single Endoflame. The armor and two greater bands give you enough regeneration to offset the mana consumption of an Invisibility Cloak.
Think of these passive mana generation options as a backup to eventually access e.g. your Hand of Ender again after somehow completely draining your mana tablets or mana rings, or if the pool your mana mirror was bound to somehow emptied. They are not meant to be your primary way of generating mana.