A check on the definition of the continuous periodogram, and does it assume ergodicity somewhere?
Eq. $(2)$ is indeed the periodogram of the truncated signal where said signal is the sample path $x(t)$ of the random process. There is nothing random about this sample path, which is why dropping the Expectation operator makes sense when one goes from $(1)$ to $(2)$. Dropping the limit is asking everyone to take your word for it that the $T$-second observation interval $\left[-\frac T2, +\frac T2\right]$ is adequate to capture enough of the general behavior of the process. Would increasing $T$ to $T^\prime &gt; T$ make a significant difference in $\hat{S}_{xx}(f)$, at least as far as the central lobe is concerned?  We don't know but we trust you to make an informed judgement.

With regard to your query $2)$ as to whether the periodogram assumes ergodicity, the answer is No and Yes.  No in the sense that Eq. $(2)$ by itself says nothing about the power spectral density $S_{xx}(f)$ all by itself: the right side of $(2)$ is the periodogram of the observed finite-length segment of the sample path of the process. Yes in the sense that it is you who can be said to be assuming ergodicity when you assert that $\hat{S}_{xx}(f)$ is a good estimate of $S_{xx}(f)$.  In general, the averages of the sample paths (e.g. the limit as $T\to \infty$ of the right side of $(2)$) need not have the same properties as the averages over ensembles (e.g. $S_{xx}(f)$), and assuming that they are the same not only means that you are assuming ergodicity (which really deals with limiting or asymptotic values of the averages) but also that $\hat{S}_{xx}(f)$ is pretty close to $\lim_{T\to\infty}\hat{S}_{xx}(f)$, that is, the short observed segment of the sample path is generally representative of the process.