In the post, you define a "meta community"1 to be healthy if it is both actively used and used for its intended purpose. I'd like to know how this notion of "meta health" is relevant for the associated site2, or in particular why it would be good for an SE site to invest in the health of their meta.
From the post, it seems you treat meta usage as an end in itself. I don't see it like that, for me meta is not an end by itself, but merely a means to an end. It is a tool to (among others) improve the quality of a site. Additionally, you talk about the intended usage. Whose intention do you mean? Different sites may have different intentions on how meta should be used, or different experiences in how meta has been useful for them. I understand that you want to have tools that are generally applicable to all sites, but I worry that you're going to reduce complex issues to a single number and miss important issues as a result.
For example, you note in your case study on Mathematics.SE that it is a healthy meta community based on usage metrics. However, I was surprised that you did not mention the following in your case study: there is significant disagreement on Mathematics.SE on how to deal with low quality questions. I'm not aware of a single place that has an overview of this issue, but discussions arising from the most recent moderator election questionairre should give you some idea. Whether this disagreement is good or bad is up for debate (and I don't intend to criticize Mathematics.SE here, so please forgive me if I misrepresent this issue), but this issue seems to be significant for that site and I think it is not good that this is issue is missed in an analysis about the "meta health" of a site.
1: As an aside, what community is this referring to? Shouldn't this just be the community formed by the site? "meta community" sounds like a group of people with the primary purpose of "being meta", which is the kind of behaviour that some argue can be a risk for a community if done in excess.
2: To be clear, I will use the word "site" to denote the group of people who visit and post on <sitename>.stackexchange.com and/or the associated meta an chat space. Some people may use the word "community" for this group, but people use this word in a way I don't understand, so I will refrain from using it myself.