Under what authority can the AIM overrule the FARs?
The issue you have appears to be that the AIM describes a case where procedures don't follow a rule (91.175) and you don't see where the authorization comes from for that deviation.  

What you're missing is that the approach procedures are in fact rules themselves.  They just don't get published in Title 14 CFR.  Standard instrument procedures are covered by Part 97, Subpart C.  It essentially states that the procedures are controlled via the FAA Order 8260 series.  

So if the approach procedure specifies minima or other procedures outside the norm of a separate rule, the procedure is the controlling rule.