Wow. Where to begin?
There's no universal answer to "Why did the Republic fall?", but I'll give you the two primary causes from my point of view.
1. The Roman patron-client system: This aspect of Roman society is often glossed over by armchair historians, but the patron-client system was the primary social institution which governed political behavior. For those that don't know, Roman patricians would essentially give money and food to plebeians in exchange for minor services and political support. The Romans were extremely socially conscious and patron-client relationships were often viewed as taking precedence over familial relations. While this was always important in Rome, the influx of poor farmers into Rome over the period of 146-133 BCE made it possible for populist patricians to suddenly gain huge swaths of clients for support in elections, etc, and the sheer speed of this amalgamation gave any moderately wealthy or ambitious senator the ability to assume a prominent role in Rome. This is what led to the meteoric rise of politicians from families which weren't traditionally powerful in the Early and Middle Republic. In fact, almost every populist that took part in the fall of the Republic came from an plebeian or obscure patrician family. In a society where the years were literally named after the consuls for that year, the impulse to gain rapid status was powerful, and the central role of the patron-client system made it possible for any decently wealthy, ambitious, or well-connected men to mold the political system to their designs. However, this wasn't the decisive problem on its own, rather the critical weakness was:
2. The imbalance of the Republican government: When the kings were overthrown and the Republic was established, the only real change was the length of terms in office. Consuls functionally had the same powers as kings, with the only real check on power being the presence of a second consul. However, there were no such limitations placed on proconsuls. Proconsuls did not have any strict term limits or formal institutional checks. Once they were in their province, the ex-consul functionally became a king, with no real limitations on their domestic powers and no real mechanism to prevent them from marching on Rome if they so choose. After 146 BCE, the Republic basically had no standing army, and every legion that was levied was (because of patron-client politics) personally loyal to the respective proconsul. At the end of the Republic, the senate only controlled Italy while each province was functionally an independent fiefdom. This flaw in the system was shone bare in 49 BCE when Caesar marched on Rome and the only thing the senate could do to stop him was flee into the arms of another powerful ex-proconsul.
The patron-client system on its own was not sufficient to bring down the Republic, nor was the executive supremacy of the consuls. But without a means to control the entire army from a central chain of command, there is no way to reign in the leverage held by proconsuls with their privately owned legions. It took the Republic 100 years to fall, but when it finally did, it stabilized under the rule of one man, and remained stable for another century to come. This was principally because Octavian was able centralize the legionary command structure. So, in my view, the only way for the Republic to survive would be to establish a large standing army in Italy, pledged to obey either the year's consuls or the senate as a whole such that proconsuls did not have the same amount of leverage. Now of course, a large military in Italy poses its own series of threats to the survival of the Republic, but so long as no one man is able to gain patronage over too many legionaries, the system might remain stable for a few more decades.