I read Roger Moore's bio and he says that because both "The Avengers" and "The Saint" did so well in the U.S. market, there was a similar expectation in ITV's projections for "The Persuaders." That's part of why the budgets and location shooting were pretty lavish.
The American ABC network did pick up the series for the U.S. after it had successfully aired in Europe. In September 1971, they put it on Saturday nights opposite "The Carol Burnett Show." It got clobbered in the ratings and moved to another night, and the ratings didn't improve. When it got canceled in Spring 1972, there were only three U.S. TV networks and the syndication market. Back then you needed at least 10 million viewers to survive on a U.S. network. Since the show wasn't getting good numbers, no one wanted it here. Since ITV couldn't justify that kind of production spending without American demand, "The Persuaders" went away. Today "The Persuaders" would find an audience on cable and do quite well, further promoted by online sharing.
As for James Bond, "Diamonds Are Forever" came out at the same time "The Persuaders" was airing. Sean Connery made it clear that was going to be his last go-around. Since Moore was then available, you know the rest.
We can Monday Morning Quarterback this forever, but ABC didn't promote this show properly. Maybe it would have worked has they used a "wheel" strategy like NBC with Columbo, "McCloud" etc. and paired it with another production. If they had dolloped out "The Persuaders" with 12 episodes a year like in Europe -- or like American cable does now with shows like "Mad Men" -- it might have run longer.
Not sure exactly where that info came from, but Wikipedia https://en.wikipedia.org/wiki/1971–72_United_States_network_television_schedule seems to show something a little different.
For part of the 1970-71 season, 'The Persuaders!' was in a really odd 9:30-10:30 slot, up against 'Medical Center', 'The NBC Mystery Movie, and 'Mannix'. For the rest of 70-71, it was against 'The Sixth Sense', and 'Mission: Impossible'.
But at no point did I see it against 'The Carol Burnett Show'.
Still, either one of those others would have been a tough slot, and frankly 9:30 - 10:30 was just really, really stupid.