Now, I recognize the thinking in using the Rings as one of the major components in the assembly of the dice pools and subsequent steps of checks, but the survey questions with their focus on approaches and especially the proportion of checks made using Rings other than a character's highest made me wonder whether this incarnation of Legend of the Five Rings might not actually benefit from dropping Rings as numeric traits altogether.
Through Stances, Approaches and Opportunities the Rings are a central part of each and every check made in the game. The choice of Ring is highly meaningfull both narratively and on a more purely mechanics level. Yet, in the current system, this choice is being confounded with the raw impact on number or dice rolled (and kept) and thus the probability of generating results. On the one hand, this could be seen as further enhancing the meaningfulness of the choices involved, by introducing an additional element of risk and a need to weigh the chance of success versus, e.g., the available options to spend Opportunities on. On the other hand, and this is were I am coming from here, this can also be seen as taking away from the inherent meaningfulness of the choice by having the yes or no nature of basic success overlay otherwise equally interesting options.
So, if we restrict Rings to just determining what you can do (e.g., the effect of Stances, the questions asked during Assessment, what you can spend Opportunities on, which checks your Anxieties trigger on, ...), the obvious question is how to determine dice pools and kept dice instead. If we want to continue using both black and white dice and a roll and keep mechanic, we need some other numeric trait to take the place of the (now non-numeric) Rings. School Rank seems like an obvious choice for me, or, if you wish to keep check results closer to what they currently are, School Rank +1 might be a slightly better fit, at least for characters straight out of character generation.
There are some more things this could potentially enable as well, but that is the basic gist of it.