Creating a metric
You want to confirm whether a metric you come up with (ratio of designer-user, designer-stakeholder interaction) is an accurate measure or approximation of UX maturity.
Define UX maturity
Figure out how to accurately measure UX maturity with an established and trusted method. To be honest, I don't know of any. MeasuringU seem to be creating one, and of course NNGroup has their definition of maturity stages.
Essentially, define what you mean by UX maturity, and figure out how to measure it properly even if its with an interview or long form survey.
Measure maturity and gather your own metric's data
Use your metric to survey designers, and also use the established method (interview or survey) with the same group.
Check for correlation
Determine the correlation coefficient of your metric vs output of the established method (like the maturity stage), and see how closely they correlate. If they highly correlate then your metric may be a good approximation of more complicated methods.
All this being said, I've never had to come up with my own metric before or validate one, but I've come to this solution because whenever I've read about usability or satisfaction measures, they all talk about how they correlate to other established measures. So it makes sense to me that the same process applies here. For example...
"Be aware that the SUS correlates strongly with a much simpler metric,
the single-question Net Promoter Score. They do provide different
data, but for many organizations, the NPS may be more useful overall,
as it’s a simpler metric to collect (one question versus SUS’s 10),
and is a well-established general bellwether for the company (even if
it’s not as sensitive to UX-focused concerns)."
Drilling down beyond a single measure
"...encapsulating all the factors into a metric [...] which is the main benefit of a metric (to 'measure' and 'improve')
rather than trying to work out exactly what the underlying factors
You're right that you need to do further research to find the underlying reason for a change in rating, but you can also ask higher quality questions to narrow it down.
A great tool I've discovered for tracking user satisfaction is this Happiness Tracking Survey from Google researchers.
The relevance here is how they use one single mandatory question (that asks you to rate your satisfaction), and then drill down a little bit by asking you to rate properties of the application (like speed, reliability etc), then drill down once again by asking about specific tasks undertaken in the application.
Do I think the metric would work?
My hypothesis is no. What if there are an insufficient number of UX designers to properly be across the work being done by a company. I may have a lot of contact with users (performing research, conducting testing sessions etc) - but user-centric design practices are not sufficiently adopted throughout the company, which to me would be a low rating of UX maturity.