I believe it's still true that you can go to YouTube and, thanks to the LBJ Library, listen to the actual party-line phone call in which Johnson had the conversation about whether to go public with Clifford, Dean Rusk, Walt Rostow, and a couple of other usual suspects. The conclusion was indeed that they needed a smoking gun that they did not have. And Johnson understood that the material was still powerful, perhaps most powerful, if he kept it close (the whole file was collected for Johnson and then put into safe keeping with Walt Rostow at the end of the administration when he left for the University of Texas.) Then Nixon would never know exactly what Johnson had and it would drive him (Nixon) crazy with paranoia. This is, in practice, exactly what it did. Alongside some more run-of-the-mill campaign sabotage activities of the old school, the real thrust of Nixon's illegal activities in the party-political realm (towards Democratic candidates and the institutional Democratic Party) were driven by fear of what might be in the Chennault file. In late spring or early summer of 1971, after he had Alexander Butterfield install the White House taping system, in one of his near-fugue states Nixon growled through a conversation about institutions of the Democratic establishment and what they might have hidden in them -- this was tied to the release of the Pentagon Papers. Nixon was sure that more PP-like documents and quite possibly the Chennault file itself were hidden away at Brookings and as he rolled through options like an actual mob boss at one point he said, "Someone needs to get in there, blow the safe." This was the single most criminal statement ever recorded on the White House Tapes -- this was conspiracy to commit burglary, not mere obstruction or vague intimations of insider dealing or bribery or campaign dirty tricks. It took until roughly the turn of the century to find because the Nixon Library, until the great Tim Naftali took over, was pretty committed to not cataloging the tapes properly and researches had also looked in the wrong place, in 1972 chasing Watergate instead of 1971. This was because, at that point, they still thought Watergate was the big thing, rather than one of several circles gobbled up whole by a much larger Venn diagram circle that encompassed everything to do with the Chennault Affair.
And of course Haldeman kept the notes. First because they all figured they could stonewall forever and no one would either ask the right question or get through various legal barriers to obtain discovery. Second because they lived every hour with the chance of legal investigation of their actions and figured (1) they needed the stuff handy in case it exculpated them from something else or could be used to blackmail their confederates and (2) so they knew just where it was in case they needed to destroy it. There were people who knew, it was a matter where you had to be smart enough to piece together the networks of people who likely knew and find the weak links, either (to compare to Watergate) feral self-interest like John Dean's or come-to-Jesus confessionals like Chuck Colson's. Madame Chennault made G. Gordon Liddy look like a scrawny little hippie, she'd have lied to the Supreme Court's face if necessary. But there were paper trails and those were problematic.
1968 was the wrong time to release for two reasons. One was the moment itself. Hubert Humphrey had made one of the great political comebacks of the 20th century in the presidential race where he had almost been trailing George Wallace on the way out of Chicago. Indeed political scientists who've read the numbers think if the election had been held the following Tuesday Humphrey would have won, both the popular vote and key marginal states. Also, with that kind of motive in the air there was no one perhaps in the history of the American republic who was better at facing the near-treasonous accusations in the Chennault file and saying, "it's all a liberal witch hunt! This is a conspiracy by that war-monger and law-breaker Lyndon Johnson to deny ordinary Americans the solutions they crave, law and order at home and peace in Vietnam!!" And he could possibly have gotten away with it. The evidence of surveillance, before you've had those crucial extra few years to learn about COINTELPRO and the intimations in the Pentagon Papers, and Sam Ervin investigating the Army spying on US civilians inside domestic borders, would look pretty damning. LBJ -- who had killed how many kids today? -- was running a secret state to ruin his political enemies just like the goddamn KGB. You really need that handful of years from '68 to Watergate, to show that Nixon himself was an even more enthusiastic and ruthless patron of a secret state designed to crush political opponents.
As a legal matter it is problematic because of the Logan Act -- Logan is frankly bad law, it's vague and circuitous and hadn't been used in so long it had practically rusted away. If you could get key documents through discovery you could certainly prosecute under Logan's terms, but there's a fair chance that either the Circuit Court of Appeals for DC or SCOTUS would toss the case because Logan is bad law, not because Nixon wasn't guilty. And then if you write a new law you get into another snarl over double jeopardy. So the actual "proper" prosecution will only ever get you limited outcomes.
But Dick Nixon himself, more than anyone else maybe even Lyndon Johnson, understood that anything to do with the Chennault Affair would never be a purely legal drama. It was the most fundamental kind of political theatre. U.S. Code, echoed in either statute or rules of evidence throughout the states, has a specific exclusion (a rule by which evidence cannot be admitted) based on "prior bad acts." Its exceptions are lethally narrow. It is intended to focus the work of trial and proving a cause of criminal action on the terms of the case itself, not questions about the defendant's character. But politics works just the opposite way. You bring this up too late -- as OTL has done, really -- and people can learn to loathe Nixon for centuries as they ought but it comes to late to do anything about him. You come at the king too early, in '68 or just after he's inaugurated, and it blows back in your face. Really the ideal time (*cough*TLplug*cough*) was if Watergate (or something very like it; there was so much the Nixon machine did that was illegal the law of large numbers alone meant they would get caught at something) blew up more fully during the '72 campaign. Then, with the Pentagon Papers and the Enemies List and rumblings of COINTELPRO and the ITT Affair (telecom giant ITT wanted a merger that made antitrust lawyers wince and paid the Nixon campaign $400K to get the Republican National Convention in San Diego where Nixon wanted it and get the AG's office to look the other way on the merger) and now Watergate-or-a-cognate, even the circumstantial evidence of LBJ's "X File" will, as the lawyers like to say, speak to a pattern of deceit. It will also bring the fire and the fury of 1970 over the Cambodia intervention and other wartime dirty tricks back into the frame, where they had been carefully tamped down by the administration in the interval.
Bring it up once the White House Tapes are out and the like and you risk Chennault's absorption in scandal fatigue. The perfect time is the '72 campaign. Probably the next-best time is right after Saigon fell, when there's a chance to write a truthful epitaph about the war, in which Vietnam was Nixon's demon lover, the driver behind nearly all the criminal malfeasance and paranoid crazy of his administration, torn between the desire to get the hell out and the soul-stalking fear that to get out and have the RVN collapse would do fatal political damage. So he lengthened the war in order to have a war he could take credit for ending, and lengthened it again to make sure Saigon didn't fall before he was re-elected. It was the prime mover. Find a prime mover and you can properly explain the universe of actions and consequences it creates. This was really the missing piece for decades, now we have it, and things sure make a more holistic kind of sense now that we have it in view. They might have then as well, but you have to time it right.