Negative log-likelihood not the same as cross-entropy?
I think the issue is that while
$$
\frac{1}{m} \sum_{i=1}^{m}\log p_{model}(\mathbf{y} | \mathbf{x} ; \boldsymbol{\theta}) \rightarrow E_{}(\log p_{model}(\mathbf{y} | \mathbf{x} ; \boldsymbol{\theta}))
$$
is true, the right side is not what is being referred to as the cross-entropy.  Indeed, we have no access to the expectation over the true population distribution / data generating process; a few paragraphs down, we find:

We can thus see maximum likelihood as an attempt to make the model distribution match the empirical distribution $\hat{p}_{data}$. Ideally, we would like to match the true data-generating distribution $p_{data}$, but we have no direct access to this distribution.

They also distinguish throughout that the expectations are over $\hat{p}_{data}$, the training data empirical distribution, not $p_{data}$, the underlying population distribution.
The left side of your limit is already an expectation, but over the finite training data, and that is what is referred to as the cross-entropy.  (&quot;Cross-entropy&quot; is a broader term, for any pair of probability distributions.  Goodfellow et al note this (my emphasis):

Any loss consisting of a negative log-likelihood is a cross-entropy between the empirical distribution deﬁned by the training set and the probability distribution deﬁned by model.


So, the answer to your questions is that the premise is incorrect: (this) cross-entropy is the same as negative log-likelihood.  Taking your questions with the limiting and population cross-entropy instead, the answer is &quot;we don't have access to the latter&quot;.  It would be the better target to be sure, but our lack of that information is the point of modeling in the first place.