The deceptively edited video that purported to show Joseph R. Biden Jr. endorsing President Trump’s re-election bounced relentlessly around the internet, falsely painting the former vice president as too confused to know what office he was running for or whom he was vying to run against.

The doctored video didn’t originate with one of the extremist sites that trade in left-bashing disinformation. It was posted on Twitter by Mr. Trump’s own social media director. From there, it collected shares, retweets and likes from the social media accounts of the president, his eldest son and the multitudinous conservative influencers and websites that carry his message to voters’ palms hour by hour, minute by minute, second by second.

The video, based on a speech Mr. Biden gave earlier this month, registered five million views in a day before his campaign responded — with statements to the press and cable interviews that largely focused on persuading Facebook to follow the example of Twitter, which had labeled the content “manipulated media.” A direct social media counterattack, aides said later, would have risked spreading the damage.

Yet the Biden camp would have been hard-pressed to mount a proportional response had it tried: Mr. Biden has only 4.6 million Twitter followers to Mr. Trump’s 75 million, 1.7 million Facebook fans to Mr. Trump’s 28 million, and nothing resembling the president’s robust ecosystem of amplifying accounts.