You've likely not realized it, but you've literally asked a version of what is probably the single-most important question to students of the Republic right now. Up there with the question of how Republican imperial expansion worked. As such I'm not going to give you an answer--if about forty years, and before that well over a century, of work by some of the brightest classicsts ever to publish can't manage it, reddit can't even begin to approach an answer--but I will give you an idea of the historiography. I've done the same thing on other questions, but your particular version is a little bit different, so I won't simply link those.

For a very long time it was thought that the plebs, urban and rural, had little to do with Roman politics. Gelzer's seminal Roman Nobility and Syme's formulation beginning in the 30s argued for a very elite, closed-off political structure, in contrast with the anachronistic model of Mommsen, who was criticized even in his own time for essentially taking his own German federal government and dropping it down on top of the Republic. From Syme and Gelzer the orthodox opinion for most of the twentieth century was that the senatorial class, and in particular the nobiles (usually defined as those with consular ancestors), controlled politics pretty much entirely internally, playing an elite game of passing magistracies and honors around. The Roman people, the granters of elected office and the source of all legislation (directly enacted through citizen assemblies at Rome), were not especially involved in the political process, and to the extent that the senatorial class actually interacted with them politically it was to secure them as clients. Masses of clients therefore dominated elections, which were not quite but almost simply for show, and could be traded around by political allies or opposed to each other. In the 1980s Fergus Millar brought up the problem that contiones, the public speeches that made up a significant portion of every Roman politician's time, don't make any sense according to this model. If the public didn't matter politically, why were Roman politicians wasting their time talking to them, often getting quite heated or even violent about it? So that I don't reinvent the wheel, a decent summary of this part of the historiography and the big names involved in this debate can be found here. Let it suffice to say that the older model--the so-called "frozen waste"--is not taken particularly seriously anymore.

More specifically, you're asking about the "need" to gain public favor, not just whether they did or didn't, or whether the Roman people held any great political power. In terms of direct political power I think everyone admits that the poorest citizens had very little legal power. They were restricted to a single century in the Centuriate Assembly, and they were predominantly a day-laborer class which would likely have had little time or energy for political activity like voting or attendance at contiones (it must be noted, however, that day laborers by definition don't get work all the time, and there is evidence that some contiones and other meetings were compensated). But what about other parts of the urban population? Meier proposed a controversial plebs contionalis, consisting of the shopkeepers around the forum who were not only in close contact with political activities but had the means to afford to be absent from their shops for a couple of days here and there. The evidence for such a group is controversial, as is the material means of urban skilled workers. Sources like the Tabula Ursonensis suggest that the skilled workers who served as magistrates' staff were not likely to be much wealthier than the day laborers who made up the capite censi. But Yakobsen pointed out that in Lily Ross Taylor's reconstruction of the property classes the requirement for the equites (HS 400k under Augustus) is fully ten times the requirement for entry into the first property class (HS 40k-50k). Could relatively affluent shopkeepers have been included in the first class? It's possible. A further problem is the depiction of the urban poor in our sources. Our literary sources are universally of educated, wealthy status. People like Cicero routinely refer to shopkeepers, slaves, and mercenarii (day workers) as being more or less equivalent--Treggiari has even shown that Cicero rarely even bothers to make a distinction between actual slaves and freedmen. But clearly a huge range of status was possible. Where should we pin political influence?

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