As ymb1 commented, RAIM was introduced in TSO-C129 receivers. These were the first GPS receivers certified and they were only approved for supplemental navigation (You were required to also have an approved nav system such as VOR as primary.)
RAIM algorithms are still present in the more modern SBAS capable receivers but are pretty much moot as long as the SBAS signal is present. SBAS provides an integrity signal that is more accurate and quicker to alarm than the RAIM algorithm.
It is also important to note that the TSO-C129 receivers were only capable of non-precision approaches (NPA). NPAs typically have an minimum descent altitude that provides approximately 500 feet ground clearance. Those limits keep you safe while using less accurate navigation systems (see also VOR and NDB.)
To address the question; when designing a system with an alerting function you have to deal with the possibility of false alerts and badly timed alerts. RAIM algorithms are not simple yes/no computations, so there's some gray area about when to alert. And losing RAIM doesn't mean you've lost acceptable accuracy. The FAA wants it to be safe but they don't want alerts that cause unnecessary go-arounds which introduce more risk and can create ATC conflicts.
Essentially there is a question of 'At what point in an approach would it be safe to continue if we lose RAIM?' You definitely wouldn't want to force a go-around 10 seconds from the missed approach point (MAP) due to loss of RAIM. For an NPA, it was determined that it was safe once you were past the FAF.
If you have reliably passed the FAF on the final approach course, you're 3-5 minutes from the MAP and there are no course changes. If you're good at the FAF, you only have to concern yourself with a potential fault that occurs between the time you pass the FAF and you reach the MAP where you either go missed or continue to landing.
The system safety assessment (SSA) addresses potential failures, their effects, and their probabilities. These are categorized by the hazards they create and whether they are detectable. The faults of concern are those that create undetected hazardously misleading information (HMI). For a NPA, the probability of undetected HMI must be less than 1E10-7 for the approach. Since approach period is 0.1 hr (6 min), the system must have an undetected HMI rate of 1E10-6/hr. (These are the same standards applied to VOR and NDB approaches.) The TSO requirements are designed to meet these safety requirements.
In the rare case of minimal satellites in view, no RAIM, and a sudden navigation fault there could be a significant change in position (and guidance) that would be detectable. With the narrowed CDI sensitivity past the FAF, any sudden position shifts that were great enough to be a hazard would generate a full-scale deflection which would trigger a go-around.