Did Jesus’s living ministry-witnesses discredit flagrantly false Christian pop-rumors (directly or indirectly)?

In the AD 30-70 Christian world, spanning across the Mediterranean regions, would the witnesses of Jesus’s life—including Mary, Peter, the apostles as a whole, and many relevant others—tend to be able to both learn of and kill off deviant rumors that were circulating about them and what they saw. In this way, would the accounts of Jesus’ life be overtly subject to public knowledge—naturally overseen, controlled, and regulated (monopolized even) by living eyewitnesses—especially the apostles—such that they had the final say on the flourishing of gospel-history traditions circulating which involved themselves? Could they thereby suppress and even weed out erroneous information if they wanted to? Could they police the development of early Christian rumors about Jesus’s life and teachings? If these witnesses were immanent protestors of rumor x, then through broader Christians echoes and conformity to witness self-disclosing, was rumor x essentially doomed to fail at being seriously circulated? At least in the earlier years, would the apostles—as known witnesses—be especially able to control the transmission of the tradition, at least one remove through direct disciples?

Remaining witnesses would exert their influence:

• Eric Eve: “It is completely fair to point out that the tradition was more likely to have been transmitted and controlled by certain individuals such as authorized teachers rather than simply passed round an anonymous collective, and it is entirely reasonable to suggest that recognized eyewitnesses were likely to have had a continuing role in this process for as long as they were around.” [Behind the Gospels (Fortress, 2014), 158 cf. 180.]
• Terence Mournet: “Within a traditional setting, eyewitnesses are a stabilizing, self-corrective force used to help counterbalance the variable character of oral communication.” [Oral Tradition and Literary Dependency (Mohr Siebeck, 2005), 187.]
• John Meier: “One would get the impression [from naive form critics] that throughout the first Christian generation there were no eyewitnesses to act as a check on fertile imaginations, no original-disciples-now-become-leaders who might exercise some control over the developing tradition, and no striking deeds and sayings of Jesus that stuck willy-nilly in people's memory.” [Marginal Jew vol 1 (), 170.]
• Stephen Patterson: “It is true, for example, that as long as people like James, Peter, or Mary Magdalene were still on the scene, the shaping and transmission of the tradition would not have been accomplished in the strict anonymity of community groups. People who actually remembered Jesus could have been around for a long time.” [“Can you Trust a Gospel?” in Journal for the Study of the Historical Jesus 6(2) (2008): 204.]
• James Dunn: “Nor should we forget the continuing role of eyewitness tradents, of those recognized from the first as apostles or otherwise authoritative bearers of the Jesus tradition" [Jesus Remembered, 242.]

Also in Diaspora:

• Michael Bird: “The problem for those who argue for widespread variation and drastic inventiveness in the Jesus tradition is that they regularly fail to reckon with the presence of eyewitnesses to the ministry of Jesus in the formative Christian communities in Palestine and even in the Diaspora.” [The Gospel of the Lord (Eerdmans, 2014), 49.]

This includes hostile witnesses:

• Craig Blomberg: “Eyewitnesses of Jesus’ ministry, including hostile ones, could easily have refuted and discredited the Christian claims during this period if they were in any way mistaken.” [Historical Reliability of the Gospels 2nd ed. (IVP Academic, 2007), 53.]
• William Lane Craig: ”Moreover, were the stories without historical foundation, the enemies of Christianity would surely have seized upon this fact.” [Assessing, 383.]
• F.F. Bruce: “[on topic xyz]…inaccuracies…would at once be exposed by those who would be only too glad to do so. ... Had there been any tendency to depart from the facts in any material respect, the possible presence of hostile witnesses in the audience would have served as a further corrective. [The New Testament Documents: Are They Reliable? (Eerdmans, 1960), 46.]

And even through AD 70 (for Gospels):

• Richard Bauckham: “Throughout the lifetime of the eyewitnesses, Christians remained interested in and aware of the ways the eyewitnesses themselves told their stories. … The period between the historical Jesus and the gospels was actually spanned, not by anonymous community transmission, but by the continuing presence and testimony of eyewitnesses who remained the authoritative sources of their traditions until their deaths. … [so they] remain an influential presence in the communities, people who could be consulted, who told their stories and whose oral accounts lay at no great distance from the textualized form the Gospels gave them” [Jesus and the Eyewitnesses (Eerdmans, 2006), 8-9.]