Analysis by Dana Bash, CNN

(CNN) -- He is ridiculed as ridiculous. He is attacked as around the bend. He is maligned as morally bereft in his unwavering defense of his old friend and client, President Donald Trump.

Yet Rudy Giuliani may just be crazy like a fox.

It turns out that he is doing exactly what the President -- his own best strategist and communications specialist -- brought him on board to do: chip away at public support for special counsel Robert Mueller's Russia probe.

It also turns out the President needed someone exactly like Giuliani to take on the role of flamethrower. His skin is thick after decades of dealing with New York tabloids, so bad press doesn't get to him as it would others.

His relationship with and understanding of the President allow him to speak fluent Trumpian. The fact that they're two Geminis in their early 70s from New York City helps make them kindred spirits.

Because of that, so far, Giuliani has avoided the fate of others, like Anthony Scaramucci, who spent a little bit too much time in the spotlight for Trump's taste.

Instead, sources in the Trump orbit tell CNN the President recognizes that the former New York mayor is helping him by taking on the role of aggressor Trump's been yearning for.

"If there is a room full of people trying to have a discussion, and there is a screaming baby, what is everyone going to think about? The screaming baby. Giuliani is the screaming baby," explained one source familiar with this strategy.

Or, put another way, the same source said, "Who is going to get the most attention? The craziest guy in the room."

An impromptu Giuliani news conference on the South Lawn of the White House on Wednesday is a perfect example.

Instead of avoiding reporters during a visit to meet with his client, as most presidential lawyers would do, Giuliani made a beeline for them, saying -- among other things -- that the part of the Mueller probe pertaining to the President must be done by September 1, so as not to meddle in this year's midterm elections.

"If he doesn't file his report by September 1, mid-September, he's clearly doing a Comey," Giuliani said, a reference to then-FBI Director James Comey's decision to tell Congress days before the 2016 presidential election that the FBI had reopened its investigation into Hillary Clinton's emails.

Taking on the Russia investigation

What is most remarkable about Giuliani's strategy is how transparent it is.

In an interview Sunday on CNN's "State of the Union" he admitted that his strategy is to erode public trust in the Mueller probe.

"[What] we are defending here is for public opinion because eventually the decision here is going to be impeach or not impeach," Giuliani told CNN. "Members of Congress, Democrat and Republican, will be informed a lot by their constituents. So, our jury -- as it should be -- is the American people."

Exhibit A: pushing what Giuliani and the President branded "spygate" -- but is really a conspiracy theory meant to undermine the Russia investigation.

Rep. Trey Gowdy, a South Carolina Republican, is spending his congressional recess trying to defuse that Trump conspiracy theory by arguing that a classified briefing convinced him the FBI did nothing wrong in the early stages of the Russia investigation.

But in the court of public opinion, the damage may already have been done.

And, in true Trump style, neither the White House nor Giuliani is backing down in their rhetoric going after the FBI -- intentionally conflating questions about a confidential source looking at Russian interference with the investigation of Clinton emails. On Wednesday, speaking to Fox News' Sean Hannity, Giuliani referred to the Mueller team as a "lynching mob."

Rough spots and a rebound

To be sure, Giuliani has faced some backlash from the President.

Not long after he joined Trump's legal team, Giuliani dropped a bomb on Hannity's show about porn star Stormy Daniels: He revealed the President had reimbursed fixer Michael Cohen for paying her hush money before the election to stay quiet about an affair she claimed she had with Trump a decade ago. (White House officials have denied her claims.)

Giuliani's remarks caught White House aides off-guard and took over the news cycle to the point where the President said Giuliani was new and didn't have his facts right.

A rough period followed, as Giuliani took beatings from legal and political experts alike.

"They should change his name to Rusty Giuliani. ... He's rusty on the law, he's rusty on the facts," Sol Wisenberg, a former deputy counsel on independent counsel Ken Starr's prosecutorial team, told Politico.

MSNBC host Joe Scarborough likened Giuliani to "a monkey throwing poo against the wall, and it doesn't usually turn out well."

Yet what looked like a mega-gaffe with Hannity turned out to be Giuliani taking a harpoon for the President.

What he knew then, but did not say, was that Trump was submitting his financial disclosure form to show the Cohen/Daniels payment.

"The strategy is to get everything wrapped up and done with this so that it doesn't take on a life of its own," Giuliani told CNN after that Hannity interview in early May.

When the disclosure finally came, it didn't make as big a splash as it otherwise would have.

Why? Because it was old news -- Giuliani had already revealed it.

When Giuliani's take-no-prisoners Trumpian tactics emerged in 2016, it was jarring for many who remembered him as America's mayor.

It is still somewhat surprising to hear the man who spent years as a US attorney working alongside counterintelligence officials calling former Director of National Intelligence James Clapper and former CIA Director John Brennan "clowns," as he did Sunday on CNN's "State of the Union."

"I prefer to remember him that way than the way he's conducting himself now," Clapper told CNN's John Berman on "New Day" Thursday morning, referring to Giuliani's leadership during the September 11 attacks.

Not long after Giuliani was hired, he insisted to CNN that what he does and says has Trump's blessing.

"You won't see daylight between me and the President," Giuliani told CNN in early May.

So far, that appears to be true.

Giuliani is doing exactly what Trump wants: Be aggressive. Shape public opinion. Take harpoons.

As with anyone in the Trump orbit, it's important to underscore that with a caveat: so far.