Dirichlet-Regression: Indicator for variable fit in model comparison

Question

I have a number of Dirichlet regression models with differing forms (log, linear, polynomial etc) over a dirichlet distribution of 6 variables/Groups.

An examplary output for summary(fit) in r would give:

------------------------------------------------------------------
Beta-Coefficients for variable no. 1: HKL0
              Estimate Std. Error z value Pr(>|z|)    
(Intercept)  5.665e-01  1.907e-01   2.971  0.00296 ** 
I(x^2)      -6.907e-06  1.334e-06  -5.180 2.22e-07 ***
I(x)         5.439e-03  1.050e-03   5.178 2.24e-07 ***
------------------------------------------------------------------
Beta-Coefficients for variable no. 2: HKL1
              Estimate Std. Error z value Pr(>|z|)   
(Intercept) -7.439e-01  2.302e-01  -3.232  0.00123 **
I(x^2)      -3.462e-06  1.536e-06  -2.255  0.02416 * 
I(x)         3.891e-03  1.246e-03   3.122  0.00180 **

[...] + 4 more variables

------------------------------------------------------------------
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

As I understood, the Pr(>|z|)-values give a Precision for each parameter of the regression model. So a low value is an indicator for a low variation and thus a good fit of that particular parameter.
(see also related but different question)

Now I want to compare how well each Variable/Group is fitted relative to the other. For example, there might be a model with an overall best fit but with high variation in one particular variable and vice versa.

So my intuition would be to compare the mean(Pr(>|z|)) over the parameters of each group. So the above example would give:

Variable 1: mean(0.00296,2.22e-07,2.24e-07) = 9.9e-04

Variable 2: mean(0.00123,0.02416,0.00180) = 9.1e-03

which tells me that Variable 1 has (nonsignificantly) less overall variation and a slightly better fit in the given model than Variable 2

But I don't want to base anything on my intuition so can anyone tell me, if i can make this comparison and why/why not? Also, what would be the correct name for such a mean?

Thanks

P.S.: any recommendation in literature and correction in my usage of statistics terminology are highly appreciated