There has been past answers on Rome/Later Han (and three kingdom) China relations from Chinese perspective diplomatic efforts by myself and u/10thousand_stars and what China knew. I don't believe we had have a Roman version yet

I can only speak to the Later Han (there were two Han dynasties, Gan Ying was of that dynasty) side. Yes they knew of each other, the Han's professional army was rather on the small side compared to Rome and one needs to take care not to draw too many comparisons though it is tempting for many. Where I am suspicious is the way the video manages to really contrast the two as if complete opposites while not entirely representing the Chinese side accurately.

I realize with only three minutes leads to simplification like Gan Ying being a formal envoy rather then Ban Chao's initiative but I would hesitate to say they viewed the Daqin as equal. As a distant land one could impose utopian ideals one, possibly Chinese themselves but had cut off from the world, a great power with much wealth but the Han was the centre of the world, the ruler the Son of Heaven and Rome's "envoys" paid tribute.

The only reference I can find in the texts (discussing Rome and Han China that I have on me) of Treasure Island is Lin Yang's work at the end when it mentions a phrase in the 4th century... the later Han control collapsed in 190 CE and was formerly abdicated by Emperor Xian to Cao Pi the founder of the dynasty in 220. I feel an error has been made by the video there

I would note that Roman's may have stoicism but there was a big appeal in restraint in Later Han China which video seems to ignore in compare and contrast. Emotional restraint, behavioural restraint and finical restraint, showing off your wealth too much risked disapproval. The Later Han sought to paint itself as more restrained then Former Han, officials were very quick to protest the Emperor against expansions of palaces, too rich clothing, tourism, harem sizes, parks. The association of the Wei dynasty with those that moved away from restraint including the libertine philosopher He Yan helped cost it support with the gentry and saw the coup of 249 that would see power leave the ruling Cao clan and go to the Sima.

Hopefully someone with knowledge of Rome can help with the wider answer, particularly the Roman perspective you seek.

Sources:

Ruler of the Treasure Country: the Image of the Roman Empire in Chinese Society from the First to the Fourth Century AD by Lin Yang

Yu Huan's Weilue in Chen Shou's Records of the Three Kingdoms, translated by Yang Zhengyuan

Fire over Luoyang:: A History of the Later Han Dynasty 23-220 AD

Chinese Historical Records and Sino-Roman Relations: A Critical Approach to Understand Problems on the Chinese Reception of the Roman Empire by Krisztina Hoppál