I can think of several, though I'll admit that these probably won't develop in the same way they did IOTL.
1. People like stability. Musical Emperors isn't going to go down well with a politically aware populace, nor is it healthy for security. At the timelines present point this isn't issue, principally because poor Emperors (distinct from terrible ones) just aren't going to be removed, but also because there is a much larger gap between the average Roman and those with political power. Presumably this gap will be much smaller in "modern" times, allowing a large number of people to grow unrestful during times of unstable government. This in itself is not an office killer.
But we're not talking about "musical emperors". The only times anything like that happened OTL were catastrophic civil war level situations.
The 13th century sees two coups (Alexius V, Michael VIII) - not counting the issue with the crusaders and Alexius IV/Isaac II.
That's why it's called absolute monarchy and not representative democracy.2. The current system gives unaccountable individuals an enormous amount of power. This is a problem distinct from issues of representation in that unaccountable "Emperor-makers" i.e. a powerful general/merchant/politician is to gain effective control over the most powerful office in the land to serve their own interests. While they may be benevolent, they may also institute an effective kleptocracy, allowing them and their supporters to run the country to their benefit. This is an issue extant in all systems of government which allow private individuals to accumulate excessive power. This is an issue which I would argue exists right now in the Empire, and always has. Ancient Rome was rife with Emperors put up by their legions, and while today's Rhomania has certainly evolved beyond this to a notable degree the laws of the land do not explicitly prevent this from occurring a la the separation of powers found in most modern countries today.
It's not 'the strongest bully". If it was, Andronicus I would have ruled twenty years instead of two (assuming old age didn't claim him) and Andronicus II would have been dropped like a PM that lost a vote of no confidence.3. From a cultural perspective, I doubt the Roman intelligentsia would be satisfied with a system which allows for the strongest bully to seize power so long as they have some kind of claim. Today we would see it is profoundly immoral that the hairiest chested man could become Emperor just because he had the muscle, as opposed to having the requisite skills to govern the nation. Nikephoros gained power through assassination; would any head of state whom was even suspected of this today be acceptable? Such barbaric practices will surely be abandoned as Rhomania advances on cultural, philosophical and ethical lines.
Which actually might be an interesting check - if the Emperor fails a vote of "no confidence', he has to step down. That might be something that doesn't undermine the monarchical element (significantly) but gives some actual "say" in things.
And those without the requisite skills to govern rarely lasted.
4. Finally, the issue of representation cannot be ignored, I don't think. While Empires in general seem set to acquire a level of legitimacy they've long since lost in our world, I strongly doubt that the same ideas of representative government won't evolve in this world. They'll certainly be different, and expressed in unusual (by our standard) ways, but ultimately educated population want to feel the government is their government. An Emperor who comes into power through the shadowy machinations of the secret service will just not have the legitimacy of one who ascends through proving to the people that he/she is capable and concerned with the interests of those he governs. Combine the former with incompetence and perhaps some extenuating circumstances and I can see a serious threat to the Imperial office looming.
Speaking as an educated member of the public and a monarchist in our world, an Emperor who comes into power because he has the Guard tagmata at his side is just as legitimate as a president who comes into office promising hope and change.
I'm not - philosophically and morally - concerned with how someone takes power, I'm concerned with how they use it.
In this world, and especially in well-entrenched monarchist Rhomania, that's not going to be an uncommon attitude, IMO.