The receiver will not sequence onto appch (or LNAV) mode if RAIM won't be available, so the idea of doing the manual check is so that you can select a different approach before you even begin, rather than waiting until it's about to transition and being unable to continue.
The argument about NAIPS' prediction being more accurate is correct, but irrelevant as it is the GNSS unit's own database (as Glekichi has pointed out) that will determine if the unit sequences onwards, not NAIPS.