Wether it survived or not
The main question is how long could it go for
Sulla was nothing more than a symptom of a disease.
Cesar simply changed the operating table to help the state function differently
Without most of these guys we wouldn't really see the same all encompassing state that rome was.
Regardless if we picture the state of the republic had it managed to survive past the imperial Era then we have to understand the means by which the late Roman Empire was able to survive and even thrive in the post Cesar/Augustus era
Romes greatest strength was the citizenship/identity of a Roman
To be Roman or The Roman Way
This gave the empire a large swathe of population to recruit from and theirfore was one of the major reasons the empire lasted so long due to a lot of different people being able to take up the Mantle of Rome and either save the state or reforge the republic into a more deadlier weapon.
We see this happen multiple acords many civil wars and crisis
3rd century crisis
Year of the 5 emperors
Tetrachy civil wars
Bagaudae rebellion
And multiple multiple others
Many of these wars and revolts getting squashed by strongman military generals who would on many occasions clash with the senate.
Seeing a rome ruled only by thr senate would've made the future prospect of Roman hegemony as Questionable
Lastly and most importantly religion/Demographics
Rome was by in large a diverse pagan state, it accepted foriegn cultures and religions in droves and assimilated them just as quickly.
But the migration crisis and rise of Christianity in Rome basically broke the concept of what exactly a Roman was
"what exactly was a Roman, A goth, a Christian Phoenician, a Greek, a Celt, a German, a syrian"
The law said a Roman citizen was by in large a Roman but the Roman state was founded by Latin Hellenised conquerors who spread civilization.
Now their were Non Latin peoples learning and spreading a Non hellenic Religion across the state built by romans.
To the ethnic Latin populations in the empire (especially in Italy)
This definitely created a revanchist ethno religious line between the identity of a Roman and a "Barbarian"
Yet alot of these questions become more pronounced as the centuries went by and the germanic threat up north became more and more pronounced while Christianity (Which was seen as a subversive threat) systematically took control of more and more of the popualtion centre's in the empire
It went from a fellowship, to a philosophy I Greece to an institution in Rome.
Now the question is asked,
"How would the republic deal with this"
We saw much of the biggest and most Heinous methods used by the romans against the Christians simply didn't work and violence wasn't stopping the spread, more subversive and hush tactics used by the state was used In the later period to create ecumenical Discord to the faith yet still Rome couldn't stop the spread of Christianity Into the rest of the empire.
The identity of what a Roman was died with the coming of the Byzantines during Justinians age, with belasarius conquest of much of the Mediterranean.
We see then the Discord that must've floated up in the germanic occupied regions of western
"Imagine being a Frankish Roman living in Gaul and a Greek/Balkan general in the east says he is taking back Roman lands from barbarians"
Taking all this into consideration I'm wondering how even after all the crisis and wars, would the romans still manage to survive the Persians or worse
"The Rise of Islam"