As a major Roger Moore fan, this premise has obvious appeal to me, but I’m not sure the “3 times” part applies.
LALD is the easy one; having failed to launch Lazenby as “the next big thing” and gone running back to Connery for DAF, UA and Eon left the impression the Bond series could never succeed without the participation of Big Tam. In addition, with the arrival of the 1970s and a very different style of moviemaking, some could be forgiven for thinking 007 belonged back in the 60s. In hindsight it’s easy to mock Lazenby for thinking Bond was a fading fad, but he didn’t make that decision in a vacuum and he was not alone in his assessment. The success of LALD proved that Bond could indeed succeed without Connery and, as secretagentfan says, ultimately that it could succeed without Moore. It would be exaggerating to say it made the world forget Connery but it did offer Exhibit A in the argument that James Bond was a phenomenon bigger than any one performer, something Cubby was intent on proving again and again in the latter half of his career. Success by success, and over time, he would do it.
It gets slightly less clear-cut with TSWLM. The lackluster performance of TMWTGG may have created the impression that LALD was a one-off, an anomaly. The fact that Harry was gone may have created some doubt whether Cubby could go it alone. The decision to double down on “Spy” was one of the boldest in series history; after three consecutive films that showed obvious signs of cost-cutting, it was a bold move to gamble on a larger budget and the construction of the world’s largest sound stage. Behind the scenes troubles in court and a mile-high stack of rejected scripts hinted at potential disaster. But the result was spectacular success and what I consider the second wave of true “Bondmania” (I may be prejudiced as I lived through it).
OP is “iffy” in my book. Yes, there was concern about a rival production, but despite McClory’s wild claims, I don’t think there was ever a way he could have spun an entire, ongoing series out of the only Bond story he owned rights to. Just owning the “SPECTRE” concept didn’t mean he owned “James Bond.” Also, even if FYEO didn’t make as much money as MR (few films did in those days!) that didn’t mean the Eon series was “on the ropes.” If NSNA had out-performed OP, it almost certainly would have led to Moore getting sacked, but I seriously doubt it would’ve ended the series, entirely.
With all that said, and taking out words like “saved,” I’m more comfortable saying something like, “Roger Moore proved Bond could thrive in a new decade and without Connery, returned the series (at least briefly) to its lofty position in popular culture and equalled or bested Sean himself to face down the threat of a rival series.”
But that’s too wordy to have any sex appeal.