When you say Aramaic was 'the language of their Empire', it was only ever so in certain contexts.

It was not the official language of the Empire in the sense that it was the language of the court, or that you would automatically expect any one Persian to speak Aramaic, or that subjects of the Empire were required to speak as a matter of course. Where Aramaic was widely used in the Empire was in administrative texts, and in certain regions as a lingua franca.

Aramaic was mostly commonly used in the Levant and Mesopotamia as a day to day language. This was due to a historical growth in its useage in these areas in a slow process after the end of the Bronze Age- the spread of Aramaic eventually displaced that of Akkadian in both Assyria and Babylonia, for example. Originally the language was associated with a specific people, the eponymous Arameans, but by the time that the Persians conquered Mesopotamia and the Levant this was no longer the case. In both the late Assyrian Empire and the Neo-Babylonian Empire that followed it Aramaic and Akkadian were used as administrative languages side by side. Importantly, Aramaic was usually rendered with the Aramaic alphabet with ink, as opposed to Akkadian cuneiform which was written into clay tablets with a stylus.

When the Persians took over Mesopotamia they inherited its administration. Akkadian continued to be used in Babylonia and Assyria specifically for local administration. Indeed, in many regions of the Empire local languages and scripts were still used for purely local affairs. But much of the Achaemenid Empire did not use the same script, and a number of areas had no immediate tradition of bureaucratic governance. We cannot talk with precision about the specific Persian decision, for we have no specific evidence about why they adopted Aramaic as a wider language. However, the strong suggestion is that its association with a swiftly-learned alphabetic script made it ideal for an expanding Imperial bureaucracy and communication network, as opposed to scripts like Akkadian cuneiform which required years to master and involved hundreds of individual characters. Key to understanding the presence of Aramaic is that it was almost exclusively used in imperial-level administrative contexts; communication between satraps and their governors, between satraps and the king, between different administrative officials, records of troop movements and of logistics.

Notably, Persia itself is an example of a region where Aramaic was only used in specific contexts, for we actually find enormous quantities of Elamite cuneiform documents in the Persian capital of Persepolis. This illustrates how Aramaic should be seen in terms of the Achaemenids specifically utilising it- it enabled communication and bureaucracy for the Empire itself, but often did not replace local languages already rendered with scripts.