Byzantium being unable to in a thousand years is inaccurate - because the Roman Empire held Syria and Egypt until the 640s, when the Arab Caliphate conquered them. So at a maximum, it's from 642-1452 - an 800 year span.

Then, from the 640s to the fall of the Abbasid Caliphate (mid 900s CE), the Byzantine Empire was dwarfed by its neighbor. Byzantine forces were always on the defensive (yearly raids into Anatolia for much of this period, and larger raids could penetrate deep into the empire to sack important cities - like Amoriam in 838, or a fleet sacking Thessalonica in 904).

So for that period of time, the Byzantines trying to conquer Syria or Egypt would not have been easy - they had smaller forces, and lost most pitched battles. It was only after the caliphate fragmented that they started to re-expand.

I would say that the period of time that Byzantium was really able to push and try to re-establish its power over the eastern mediterranean would be from the mid 950s to the death of Basil II ( 1025), or during the Komnenian Restoration (1080s to 1180s). But after the sack of Constantinople in the 1200s, the empire was fatally weakened - even after taking the city back, it would never really be able to become ascendant again.

So really, to rephrase your question - the Byzantines, from (whenever you consider Byzantium to start) to 642 owned Syria and Egypt. From 642-~950, they were too weak to defeat and conquer land from the massive caliphate. And from the sack of Constantinople in 1204 onward, they were too weak to reconquer Anatolia fully. That leaves the period from 950ish to 1180ish as the only real opportunity to reverse the 'decline' - not a 1000 years.

In comparison, by 1453 the Ottomans were on the rise. Conquest is not necessarily always slow - for example, the Romans conquered Syria over a 10 year period under Pompey, and Egypt over a few years of civil war. They were lost twice - once to the Persians, over a 20 year period, and once to the Arabs, over a ten year period. A state winning battles, with a powerful military and competent ruler, can conquer large areas of land quickly - which is what the Romans did in the East from 70BCE-10BCE, the Arabs did in the 630s/640s CE, and the Ottomans went on to do before and after the conquest of Constantinople.