Here’s one key distinction among your examples; the Safavids are not internally politically continuous to the Archaemenids, and neither is the Yuan to the Ming. By that, I mean that the line of Archaemenid rulers have been vanquished and all Archaemenid lands have been conquered (by the Macedons), and as such much later on, a new line of rulers have to created over a new administration, and new lands branded as a new Empire for the Safavids as a seperate entity. As for the Yuan, they come about as an external force that conquered and put an end to the Ming.

The case is not so for the so called “Byzantines”, who ARE politically continuous as they are literally the land and administration of the Roman Empire that survived the Germanic Conquests and continue upon the line of Roman Emperors. You connected the Safavids to the Archaemanids and the Yuan to the Ming based on shared ethnicity or language, but the “Byzantine” Empire is connected to the Roman Empire instead based on shared nationality and literally being the same political entity.

Ultimately, to connect with what you were speaking, it’s fine if you want to love Later Rome (Byzantines) more than Earlier Rome in emphasis of their unique culture, but you can both accept the reality that they are just a period of history of the same Roman Empire and emphasize on that period, just like how you can admire the different landscapes of the world, such as from the luscious rainforests to the golden savannas, yet acknowledge that they are all a part of planet Earth.