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I also dug up an old post of mine addressing a very similar subject, so here's some additional thoughts, if you were curious:
The Republic surviving seems like a stretch to me. Sulla tried the whole "reform the constitution" angle, but Pompey and Crassus had reduced all of Sulla's institutional work to shambles with the brute force of the army. You mention "military reform", but the military "expressing itself" was not the problem. Every legionary was (officially) a Roman citizen, so they could already all vote, and unrest for the political rights of the legions was not what destroyed the Republic. The institutional problem at the root of everything (at least according to one view on the fall of the Republic) was that the governing framework of the Republic was only equipped to deal with the government of a city state. It was able to adapt to governing Italy effectively (after the Social War at least), but when it came to the rest of the provinces, the senate basically sent out one of their own and said, "Just do whatever for a few years then come back." The Romans were so concerned about having checks and balances in Italy (with two consuls, ten tribunes, senatorial bylaws, and three different electoral assemblies), but they basically gave unlimited absolute power to provincial governors with absolutely no checks on their authority. This can be seen most plainly in the year 49 BCE when the senate tried to recall Caesar, but the only thing the senate could do when Caesar marched south as a warlord was turn to another warlord for protection. That display on its own was proof that the provincial governing system was broken because they weren't even really under the control of the senate. Each province was basically an independent private fiefdom led by whatever governor was in charge. Sure, they were usually loyal to the senate, but it only took a few exceptions to bring the entire Republic to its knees.
The only way to save the Republic, imho, would be the total reform of the provincial government system. Caesar himself tried to tackle this during his consulship in 59 BCE with the lex Julia, which tried to impose penalties for corruption, bribery, etc in the provinces, but it failed as a reform measure for the Republic because there was no means by which the government in Rome could hold governors accountable. And this was by design, the Roman concept of military command was centered on the idea of imperium: that a commander has unlimited power of life and death over his soldiers and answers to no one. The Romans countered the possibility of corruption by limiting consular and proconsular terms to one year, but by the late Republic generals started raising private armies to blackmail the senate into backing their agendas. The senate couldn't do anything but watch and acquiesce to the generals' will because the senate itself didn't have an army to stop them. Anytime the Republic underwent civil wars, there was never an "army of the Republic", there were only armies of Sulla or Cinna or Pompey or Caesar or Octavian. To stop this from happening, the senate would have to do one of two things:
1. Create a unified military command: This is essentially what Augustus did to stop the chaos of the civil wars. He took all the provincial legions (each run by their own independent warlords) and put them all under a single commanding officer (himself). Later emperors would reform this system by adding additional general-level officers to make command of the legions more efficient (duces, magistri milita, magistri equita, etc). Of course, this just moves the underlying problem one step up. How could the senate hold this unified command accountable or check their power? Would they place the consuls at the top of this command? The office of consulship itself was frequently awarded to men by means of bribery or political violence, so I doubt that would be a reasonable measure, so the Republic may have to create a whole new office to command this new military model, but Caesar got assassinated for doing exactly that, so its doubtful that the institutional inertia of Rome would allow for such circumstances to arise.
2. Redefine the legal basis of imperium: Giving singular provincial governors total imperium in the provinces was, as I have said, part of the underlying problem. It would be like the US constitution giving US states the power to raise their own armies and didn't make state governors abide by the constitution or any federal laws, and then abolishing the US military. It would be chaos. The senate would have to start either sending two or three governors to each province (which was done during the empire), perhaps giving them different responsibilities. For example, the proconsul might command the legions and raise taxes, but the procurator might be the one who collects those taxes and pays the legionaries, so neither one would be able to turn the legions on the other and take over absolute control of the province (at least in theory). Or perhaps you could have two equal proconsuls who each command the province for a month (as they had in Rome).
I doubt either of these solutions would stick, since they both imply rapid and sweeping institutional changes to the Roman government, which were never very popular, since the Romans valued tradition so heavily. That's why from the founding of the Republic to its fall, the only significant institutional changes were the concessions given to the plebeians (the tribunary veto, plebeian consuls, the right to serve as legionaries) and the ad hoc creation of the province system.