In Humanae Vitae, after the Pope reaffirmed the ban on contraception, he urged scientists to continue researching and improving natural methods. Some Catholic scientists and Catholic universities (Creighton, Marquette, Georgetown, etc.) took him up on it. This is not to belittle non-Catholic contributions, but that's a big reason why so much of the research is from Catholic sources.
(FWIW, the Catholic Church does not consider FABMs to be contraception, but "well-timed abstinence". Some of the Catholic sources can be really confusing if you're not aware of this terminology.)
The 24% failure rate cited by Planned Parenthood includes ALL couples who claimed to be using fertility awareness, from those who had taken classes and were using a modern method, to those who were just guessing. IIRC, over 80% of the couples were not properly trained in a modern method of fertility awareness. Not surprisingly, they had very high pregnancy rates. Someone did the math (can't find the article) and figured untrained users had about a 28% pregnancy rate while trained users had a 7% pregnancy rate. (80 * .28 + 20 * .07) = 23.8 The 7% pregnancy rate is comparable to the real-world use of the Pill.
As for your system analogy, if your automated system was a significant resource hog or ran the risk of corruption of data or crashing the system, would you use it? You focus on efficacy without factoring in side effects.
(FWIW, the Catholic Church does not consider FABMs to be contraception, but "well-timed abstinence". Some of the Catholic sources can be really confusing if you're not aware of this terminology.)