My understanding was that the constitution of the Republic put all power in the hands of the Legislative Assemblies and their designated representatives, the magistrates and tribunes. The Senate, as it happened, was not considered one of the Legislative Assemblies; the Assemblies consisted of the Centuries (based on the old military levy of Rome), the Tribal Assembly and the Plebian Assembly. The Centuries elected all the magistrates except the tribunes, who were elected by the plebes. The Tribal assembly passed most legislation, thought the Plebian Assembly could also legislate.
The Senate was originally formed by the Kings of Rome as a consultative body; it had no formal legislative powers. However, the Senate consisted of the wealthiest and most influential men in Rome, as well as the leaders of all the major families. These men also had the leisure time to attend to politics, and so were usually the best-informed and best-connected men. The Senate met regularly (unlike the Assemblies) and so had practical experience as well. In short, the Senate wielded a great deal of influence, and if it was united it effectively ran the city. The existance of the other Assemblies only mattered if the Senate was split into factions. It became increasingly so after the Punic Wars, and Roman politics became more violent and intense, eventually collapsing the Republic.
The Gracchi brothers had the right idea exactly in their reforms. By the time of their emergance, the Roman economy was dominated by immensely wealthy landholders using slave labor, driving out the smallholder that had made the Republic strong. Most of the large landholdings were not private property, but state land siezed during the Punic Wars and leased at bargain rates to influential senators. The Gracchi proposed breaking up the state lands to restore the smallholder farmers. The wealthy landholders objected for obvious reasons, but the Gracchi program was so popular that land reform legislation could not be stopped in the Assemblies. So the Gracchi were murdered.