Combine that with the rise of a new powerful enemy and you have a disaster on your hands.
New evidence suggest that the Theme system was only barley function for a long time before the decline. In fact, going away from that system is partly why they were so powerful by the year 1000.
> by the end of Heraclius's [1] reign
I don't agree with that at all. If there is any reasonable point to talk about non-continuation it would be 1204.
I stand by my opinion regarding the Heraclius era. I would have said Justinian, but one can still see traces of the Roman heritage here and there during his reign, but after Danube limes fell in the late 500s it wasn't the "hellenised" Romans battling the Slavs and Avars, but the Byzantines (yes, I know they were calling themselves the Rhōmaîoi).
Back to Heraclius's reign, once the Arabs came on the scene things were totally, totally different. I know people now like to diss on Pirenne for his Mohammed and Charlemagne and for how he insisted on the disruption, in fact the total destruction, of most of Mediterranean trade caused by their advent on the historical scene, but he was, in essence, right, especially when it comes to the "Western" Mediterranean (not sure how I would exactly define "Western" Mediterranean for that period, hence the quotes). Things were never the same after that. Also, the Byzantine Iconoclasm phenomenon wouldn't have happened without the close contacts with the Arabs, and, again, this is of course my own personal pov, Iconoclasm is totally a non-Roman-like phenomenon.
And before someone starts saying that the Roman Empire lived "through" the Byzantine Empire all the way to 1453 I have to say that from my point of view that is only valid until the late 6th-early 7th century, by the end of Heraclius's [1] reign the Byzantine Empire was a thing totally different compared to the Roman Empire.
[1] https://en.wikipedia.org/wiki/Heraclius