This is a partial answer, for the record, and hopefully somehow helpful.
Remark 1. The multi-lag empirical covariance matrix $\widehat{C}$ in your question may contain parametric information about the underlying dynamical law generating the time series -- what information will depend on the dynamics itself.
For simplicity, in what follows, I will consider the exact covariance $C = \mathbb[X X^{\top}]$ computed with the expected value, instead of the empirical one $\widehat{C} = \frac{1}{T+1}\sum_{\ell = 0}^{T} \mathbf{x}_{\ell} \mathbf{x}_{\ell}^{\top}$ computed with a finite sample mean.
1D Linear Dynamical System. More concretely, consider a discrete-time linear stochastic dynamical system $$ x[n+1] = \rho x[n] + \xi[n+1] $$ with $\left|\rho\right|<1$; $(\xi[n])_n$ i.i.d. with finite moments and zero mean; and $x[n]$ independent of $\xi[n]$ for all $n$. Assume that the process $(x[n])$ is initialized at the invariant distribution. Then,
$$ C_{i,i+k} = \mathbb{E}\left(x[i]x[i+k]\right) =: R_k = \rho^k R_0, $$
where $R_0 = \sum_{\ell=0}^{\infty} \rho^{2\ell} = 1/(1-\rho^2)$ is the correlation and $C_{i,i+k}$ stands for the $(i,i+k)$ entry of the (limit) covariance matrix $C$, namely, it is the $k$-lag correlation.
These yield two useful estimators for the main parameter $\rho$
$$ \frac{C_{i,i+1}}{R_0} = R_1 (R_0)^{-1} = \rho \,\,\,\,\,\,\,\,\,{\sf (Granger)}$$ $$ C_{i,i+1} - C_{i,i+3} = R_1 - R_3 = \sum_{\ell=0}^{\infty} \rho^{2\ell+1}-\sum_{\ell=0}^{\infty} \rho^{2\ell+3} = \rho, $$
where the latter identity holds from telescopy. Indeed, $\rho$ is a key parameter for the qualitative properties of the linear dynamics.
Multivariate Linear Dynamical System. In this case, the dynamics is given by the counter-part $$ x[n+1] = A x[n] + \xi[n+1] $$ where $A$ is an $N\times N$ symmetric stable matrix whose support entails the graph of interactions among the nodes $m=1, 2, \ldots, N$.
The same estimators extend to this setting $$ C_{i,i+1} (R_0)^{-1}= R_1 (R_0)^{-1} = A \,\,\,\,\,\,\,\,\,{\sf (Granger)}$$ $$ C_{i,i+1} - C_{i,i+3} = R_1 - R_3 = A, \,\,\,\,\,\,\,\,\, (\star \star)$$ where now $C_{i,i+k} = R_k$ is a matrix: The $k$-lag covariance matrix.
Unveiling the graph of interactions. Assuming that $A$ is nonnegative: Nodes $i$ and $j$ are connected if $A_{ij}>0$, otherwise, if $A_{ij}=0$, then they are disconnected. With this in mind and in view of the identity $(\star \star)$, we have that the set of vectors $\left\{F_{ij}\,:\,i,j = 1,2,\ldots,N\right\}$ defined as
$$F_{ij} \overset{\Delta}= \left(\left[R_1\right]_{ij},\left[R_2\right]_{ij},\ldots,\left[R_M\right]_{ij}\right)$$
for $M>3$ is consistently linearly separable, i.e., there exists a hyperplane that separates the vectors associated with connected pairs $\left\{F_{ij}\,:\,i\sim j\right\}$ from the vectors of disconnected pairs $\left\{F_{ij}\,:\,i\nsim j\right\}$. This means that $C$ entails information about the underlying graph of interactions: To recover, you need to properly cluster the set $\left\{F_{ij}\right\}$.
Remark 2. The parametric and structural information entailed in $C$ and discussed rely on the linear dynamics underlying the time-series. For other dynamical laws, the guarantees should be reworked accordingly.
P.S.-- My apologies: You asked something (how the spectra of the empirical covariance $\widehat{C}$ universally informs about the qualitative properties of the system) and I answered something else (how the exact $C$ itself can inform about certain built-ins of a linear dynamical law). Perhaps someone can provide a more complete answer.