Why do we care more about test error than expected test error in Machine Learning? In Section 7.2 of Hastie, Tibshirani, and Friedman (2013) The Elements of Statistic Learning, we have the target variable $Y$, and a prediction model $\hat{f}(X)$ that has been estimated from a training set $\mathcal{T} = \{Y_1, ..., Y_N, X_1, ..., X_N\}$. The loss is denoted $L(Y, \hat{f}(X))$, and then the authors define the test error:
\begin{equation}
\mathrm{Err}_{\mathcal{T}} = \mathbb{E} \left[ L(Y, \hat{f}(X)) | \mathcal{T} \right]  ,
\end{equation}
and the expected test error:
\begin{equation}
\mathrm{Err} = \mathbb{E} (\mathrm{Err}_{\mathcal{T}}) .
\end{equation}
The authors then state:

Estimation of $\mathrm{Err}_{\mathcal{T}}$ will be our goal...

My question: Why do we care more about $\mathrm{Err}_{\mathcal{T}}$ than $\mathrm{Err}$?
I would have thought that the quantity that measures expected loss, regardless of the training sample used, would be more interesting than the expected loss that conditions on one specific training sample. What am I missing here?
Also, I've read this answer here which (based on my possibly incorrect reading) seems to agree with me that $\mathrm{Err}$ is the quantity of interest, but suggests that we often talk about $\mathrm{Err}_{\mathcal{T}}$ because it can be estimated by cross-validation. But this seems to contradict Section 7.12 of the textbook, which (again by my possibly incorrect reading) seems to suggest that cross-validation provides a better estimate of $\mathrm{Err}$ than $\mathrm{Err}_{\mathcal{T}}$.
I'm going around in circles on this one so thought I would ask here.

+1 to Demetri Pananos's answer.
It may well be that we apply the same model $f$ to two different training datasets $\mathcal{T}$ and $\mathcal{T}'$. And $\mathrm{Err}_{\mathcal{T}}$ may be quite different than $\mathrm{Err}_{\mathcal{T}'}$ - either much lower, or much higher. This may be of vastly larger importance when we actually apply $f$ than the expected error $\mathrm{Err}$ over all possible $\mathcal{T}$s.
As an example, I do forecasting for supermarket replenishment and apply my model to many, many training datasets (essentially, historical sales of one product at one store). The loss directly transforms into the necessary safety stock. It's much more important to know the necessary safety stock per product and store than the "overall" safety stock.
