Why cannot the model $\frac {y_{i,j }} {N_{i,j } } = \beta_0 + \beta_1 X_i + e_{i,j }, \ y_{i,j}\sim B(N_{i,j},\pi_i)$ have constant variance? The following example is taken from a book by Walter Stroup on Generalized linear mixed models, and are supposed to show some limitations on trying to write models in equation form.
Let $y_{i,j } \sim$Binomial$(N_{i,j},\pi_i)$ and suppose that the proability $\pi_i$ of success depends on the independent random variable $X$. We would like to model the change in the probability of a success associated with changes in $X$.
Now quoting from the book, where $e_{i,j }$ are i.i.d. $N(0,\rho)$.

We redefine our response variable as $p_{i,j}  = y_{i,j} /N_{i,j} $, the sample proportion for the $i,j$:th
  individual. This gives us $p_{i,j}  = \beta_0 + \beta_1X_i + e_{i,j} $ as a candidate model. Assuming sufficient $N_{i,j} $ for
  each individual, we could invoke the central limit theorem and claim $p_{i,j} $ have an approximate
  Gaussian distribution. However, unless $\beta_1 = 0$—that is, identical $\pi_i$ for all $X_i$—we cannot
  assume constant variance $\sigma^2$.

I'm not able to understand the bold part. If we assume $X_i$ and $e_{i,j }$ are independent then the variance of $p_{i,j } $ is the sum of their variances, which would be constant if $X_i$:s are distributed with the same variance. Is this something totally undesirable or is there something else which make it impossible for the variance to be constant? 

I should also add that the author does not claim that this is the only reason moedls in equation form are undesirable but also gives other reason.
Grateful for any help provided!
 A: Firstly, this is a very strange model --- it is a binomial GLM with an identity link function, which is a strange link function to choose in this context.  In any case, even taking that model as fixed, the explanation in the book seems quite strange to me.  Re-arranging the regression equation, and taking the variance of both sides (treating the explanatory variable as a constant), you obtain the error variance:
$$\begin{equation} \begin{aligned}
\text{V}(i,j) \equiv
\mathbb{V}(e_{i,j}) 
&= \mathbb{V} \bigg( \frac{y_{i,j}}{N_{i,j}} - \beta_0 - \beta_1 x_i \bigg) \\[6pt]
&= \mathbb{V} \bigg( \frac{y_{i,j}}{N_{i,j}} \bigg) \\[6pt]
&= \frac{1}{N_{i,j}^2} \cdot \mathbb{V} (y_{i,j}) \\[6pt]
&= \frac{1}{N_{i,j}^2} \cdot N_{i,j} \pi_i (1-\pi_i) \\[6pt]
&= \frac{\pi_i (1-\pi_i)}{N_{i,j}}. \\[6pt]
\end{aligned} \end{equation}$$
This variance function will only be constant if $N_{i,j} \propto \pi_i (1-\pi_i)$ for all $i$ and $j$, which would be extremely unusual and specific requirement (especially since the count values would all be integers).  Even if $\pi_i$ were constant for all $i$, this would not be sufficient for constant variance, since the higher count values would still diminish the variance of the error term.  So, unless there is some other aspect of the explanation in the book that is not made clear in your question, I'm afraid I can't see any sense in the quoted section.
