I mean the problem is not necessarily exclusive with Autocracies/monarchies. The Roman Republic had basically a century of anarchy and instability from years of bad leadership which led to Augustus. He was an autocrat, but after him no one wanted to go back to the older system.
That is definitely true and one of my favorite realizations in my studies of Roman history was that the fall of the Roman Senate was a largely popular revolution against tyrannical rule. (Edit: I should probably clarify that I don't mean that it was a popular revolt as in the French Revolution, but popular as in the average people were generally supportive of the move.)
Though I do remember that at least in part, even when offered the authority the Senate wouldn't take it back after a certain point, having grown used to being a social club with no real responsibility. Though I could be misremembering. It has been a while.
This already was in practice. The aristocracy, the bureaucracy, and the army served as checks to the Emperor. The Emperor's power while more absolute than most of his peers in Europe was not unquestionable. He needed the support of the facets of government and Roman society. After all many Patriarchs have made or broken Emperors. The army for one preferred peaceful hereditary succession. It made the line of succession clear and ensured that there was some stability.
The problem with the Empire was that it was always dealing with various enemies on the frontier which left it highly militarized. This necessitated soldier-emperors to take throne, so any form of weakness, or passivity was really looked down upon. If you have a dynasty that occupies the throne long enough, the matter of succession could be smoothed out as various dynasties like the Macedonians or the Heraclians came quite close to this. The Empire had over a century of peaceful succession actually going back to Anastasius which was undone by Phocas' idiocy.
I've wondered if stability through hereditary succession was the wrong goal. The stability was definitely critical, but what I wonder is if there could have been a way to limit the Emperor's powers primarily to military concerns, and placing a civilian authority in charge of other concerns with the idea of both making the seat of the Emperor somewhat less desirable while retaining the benefits of central military leadership.
This way, they could theoretically treat the office of Emperor as a non-hereditary military office, hopefully just appointing capable generals in the position, perhaps as coemperor to ensure someone capable is always leading the defense, and a more dedicated civil authority runs things on the home front.
I'm just spitballing ideas right now because I don't have access to any of my books right now, but I mostly want to try to think of a way limit the violent civil wars and occasional complete incompetent in the Empire's leadership, while retaining the Empire's remarkable ability to find capable leaders in times of crisis. I feel like a true hereditary succession would break that.
This also is a double-edged sword as it could prevent the Emperor from implementing necessary reforms. The Kings of France found out this lesson as the Ancien Regime experienced growing pains as it tried to consolidate and expand. This led to a lot of conflicts with the nobles and long periods of reform which spiraled into the Revolution
That is fair. There is potential for disaster I suppose in any system. Nothing is perfect after all, but it would be very nice to limit the potential damage of poor leadership, even at the cost of slower reforms.
I think the problem here is a serious case of presentist thinking.
"Byzantium fell, and these nations with radically different circumstances didn't, ergo Byzantium had to be like these nations even when the situations aren't the same or even applicable."
It's like saying "Every nation should be like the USA or China because these states are the most powerful right now."
and then you get weird stuff like making Rome just the USA with serial numbers filed off.
I would never try to apply US ideals to the Eastern Romans. They are coming from completely different circumstances and traditions.
That's why I wasn't very specific with my ideas. I don't really know for sure exactly what would work for their situation, I am just reasonably certain that in order for the Empire to hold Southern Italy, and prevent catastrophic collapses in general, some meaningful changes to their system of governance and especially succession would be necessary.