So, in the case of the Achaemenids you have a basic issue with sources. In the case of India, they got a writing system for Sanskrit due to the Achaemenids, toward the end of the dynasty. Brahmi script is derived from Imperial Aramaic script and there are no historical records before the edicts of Ashoka, well after the fall of the Achaemenids - and even after that I believe they are very sparse. Similarly, the united predecessor to "China" as we think of it would only emerge under Qin Shi Huangdi in the 3rd century, and the first attestation of interaction with Central Asia comes from Sima Qian around 100 BC - and they're very terse.

Indeed, what we see is that outside Greek sources, the Achaemenids - bar Darius III - were largely forgotten (a vague memory of Artaxerxes II can also be found in the Shahnameh, if I recall correctly). In Central Asia, dynasties would trace their heritage not to the Achaemenids but to Arsaces, founder of the Parthian Empire (though one comment in a western source, that he took the name Arsaces after Artaxerxes I or II's birthname, makes me wonder if he was conflated with these very long-reigning and significant Achaemenid kings). There is a tradition that connects the Sasanian dynasty to Darius III (Karnamag-i Ardašir), but it is unclear how early it was or whether it emerged due to 4th-5th century interactions with the Greeks; royal inscriptions merely make vague mentions of notoriously cryptic "ahenag/hasenag", or 'forebears'. A common interpretation these days is that these were imagined to be the Kayanids, a mythologized version of the Achaemenids.

In any case, I would be extremely surprised to find any unique perspectives simply due to the lack of attestation, or in the case of China, even interaction, that early. There are definitely some Chinese records of interaction with the Parthians/Arsacids and Sasanians, but I don't know these off-hand, and I don't know what Indian sources in those eras look like. But with the Achaemenids, I believe you are out of luck.