Outside of Rome, the military usually enforced the edicts of magistrates and the rule of law. However, no weapons could be carried within Rome’s boundaries, the pomerium. In the absence of this ordinary method of policing, the Senate came up with the senatus consultum ultimum (SCU) in 121 BC, when Gaius Gracchus was causing unrest (Plutarch, Life of C. Gracchus 14.4). This decree ordered all magistrates to take up arms in defence of the State, as Plutarch describes, and seems to have superseded the ordinary limit on weapons being carried within the pomerium, enabling the magistrates to act as a civilian police force.
For example, Sallust tells us that praetors arrested Catiline’s allies in 63 BC, when Catiline allegedly conspired to overthrow the Republic (Sallust, The War With Catiline 45) - through the SCU, the magistrates became a de facto police force. Interestingly, the magistrate’s guards are also described, which perhaps suggests that the magistrate’s authority was passed on to them as a result of the SCU.
To draw this back to your question, the answer is that the Republic had the capacity to form a police force when necessary. With regard to your example of the riots, the nature of the SCU meant that it had to be passed in the Senate. It seems likely, therefore, that a certain proportion of the Senate were not opposed to the riots, and so did not pass a SCU. Of course, were the Senate to create a police force independent of political interest, that would only weaken their position and be of no benefit to them.