Without getting into a discussion of the problematic concept of feudalism, the "feudal system" relies on interpersonal relationships and familial power dynamics in order to maintain control of land. The ruler, whether it be a king, duke, gastald or any other inflection of landed office, requires that his "vassals" are loyal and the land under his nominal control is inherited by the correct people. The only definition of "overstretched" that matters is how effectively a ruler could keep his vassals in line and biddable. This is also tied to the monetary systems of the medieval period - rarely did kingdoms effectively exercise taxation without undermining the interpersonal relations that held it together. Without tax, relying on the goodwill of the landholders, feudal states couldn't maintain standing armies. As you pointed out, you cannot hold land without armies. The Roman Empire had legions and auxiliaries to be stationed and garrisoned where needed. The Byzantine Empire had tagmata and themata which fulfilled the same purpose. In short, holding land requires the means to supply, pay and equip armies. This in turn comes down to possessing a tax system, which most feudal systems lacked.

Imperial systems such as practiced by the Roman Empire and indeed the Byzantine Empire relied on bureaucracy to administer land held by the state. Furthermore, the Roman Empire and the Byzantine Empire had established and functioning taxation systems. These systems had flaws and periods of intense abuse and corruption which impacted their effectiveness, but for the most part, the Roman governors and the Byzantine strategoi did not own the land under their control; they simply managed it on behalf of the state. To stress the point: the Byzantine Empire was not feudal in any sense of the word. Anthony Kaldellis wrote a brilliant critique of the concept of 'Anatolian magnates' in his book Stream of Gold, Rivers of Blood which is worth paraphrasing: "no evidence for provincial power bases".

The Byzantine Empire losing control of Italy comes to a flaw of the bureaucracy being unable to keep up with the extreme level of external pressure in the form of the Lombard conquests. Pope Gregory I wrote many letters to the Byzantine emperor Maurikios which reveal to us that the troops stationed in Italy went unpaid for long periods of time and were of insufficient quantity to fight the Lombards. There is also the possibility that Romanos, the exarch of Italy, was misappropriating imperial resources at the time. Furthermore, Italy at this time was 'worth' very little to the Byzantine state. Ravaged by the Gothic Wars of the 6th century, with Lombard states established in Spoleto and Benevento, Italy was probably not worth the expense required to hold onto it.

In summary: feudal systems are not "limited from the start" by any number of vassals - they could be enormous such as the Angevin Empire. They are limited by their very structure, which effectively prohibited effective taxation and forced rulers to rely on the loyalty of their vassals to maintain power and control land. Empires had a large state apparatus which made them susceptible to corruption and malfeasance, but much more able to consistently hold land.