Model selection and assessment using leave-one-out cross validation

Assume $D$ is the training data set with both the value of the predictors $mathbf{X}$ and the value of the response variable $Y$. I have a loss function $L$ and two models $f(mathbf{X};beta)$ and $g(mathbf{X};lambda)$, where $beta$ and $lambda$ are model parameters. Our goal is to estimate

begin{equation}
e(f)=mathbb{E}[L(Y,f(mathbf{X};beta))|D]~mbox{and } e(g)=mathbb{E}[L(Y,g(mathbf{X};lambda))|D]
end{equation}
Note that it is the expectation of the generalization error of the model $f$ (and $g$) that is trained on the specific training data set $D$, where the expectation is taken based on the same distribution that generates $D$.

Now, if we do a leave-one-out procedure, specifically: let $N$ be the total number of observations in $D$, let $D_{-j}$ be the data set that removes the $j^{th}$ observation. Then,$L(Y_j,f(mathbf{X};beta))|D_{-j}$ should be an “almost” unbiased estimator of $e(f)$ right? Theoretically, to get $e(f)$, once should generate infinite new $(Y_i,mathbf{X}^{(i)})$ from the distribution of $D$, then train the model on $D$ and use that model to make prediction on the new infinite data set and take the average. Now we fit the model on $D_{-j}$, which is only slightly different from $D$. So $L(Y_j,f(mathbf{X};beta))|D_{-j}$ should be an “almost” unbiased estimator of $e(f)$. Then you go through all $N$ data points in $D$ to obtain the value of $L(Y_j,f(mathbf{X};beta))|D_{-j}$, where $j=1,2,…,N$ (assume $N$ is large), then you take the average, then the result should be very close to $e(f)$ right? Then we do the same thing to model $g$. Then in this case, we get very good estimates of $e(f)$ and $e(g)$ so we can do model selection based on $e(f)$ and $e(g)$. Specifically, if $e(f)<e(g)$, then I should expect that on a LARGE new independent data set $T$, $f$ should perform better than $g$ correct? Also, the quantity $e(f)$ and $e(g)$ computed from $T$ should be very close to the quantity computed from $D$, assuming both $D$ and $T$ are large. Is that correct?

If all the above are correct, then it seems that I did model selection and model assessment in one step. But should I partition the data set into 3 pieces, i.e., first train 2 models on one piece, then apply the 2 models on another piece to do model selection, then apply the 2 models on the 3rd piece to do model assessment?