This question is about definition 17.12 (of the strong markov property) and the following remark 17.13 in A. Klenke's Probability Theory.
For completeness, the definition: Let $E$ a polish space, and let $I\subset[0, \infty)$ be closed under addition. A Markov process $\left(X_t\right)_{t \in I}$ with values in $E$ and distributions $\left(ℙ_x, x \in E\right.$ ) has the strong Markov property if, for every a.s. finite stopping time $\tau$, every bounded $\mathcal{B}(E)^{\otimes I}$-$\mathcal{B}(\mathbb{R})$ measurable function $f: E^I \rightarrow \mathbb{R}$ and every $x \in E$, we have \begin{equation*} _x\left[f\left(\left(X_{\tau+t}\right)_{t \in I}\right) \mid \mathcal{F}_\tau\right]=_{X_\tau}[f(X)]:=\int_{E^I} \kappa\left(X_\tau, d y\right) f(y) . \tag{$*$} \end{equation*} Here, $κ: E \times \mathcal B(E)^{⊗ I} → [0,1]$ is defined by $κ(x,A) = ℙ_x[X ∈ A]$, which is a stochastic kernel by definition of a Markov process.
The remark following this definition states that if $I$ is countable, then the strong Markov property holds if and only if, for every almost surely finite stopping time $\tau$, we have \begin{equation*} \mathcal{L}_x\left[\left(X_{\tau+t}\right)_{t \in I} \mid \mathcal{F}_\tau\right]=\mathcal{L}_{X_\tau}\left[\left(X_t\right)_{t \in I}\right]:=\kappa\left(X_\tau, \cdot\right) . \tag{$\diamond$} \end{equation*} Here, $\mathcal L_x[(X_{\tau+t})_{t \in I} | \mathcal F_τ]$ denotes a regular conditional distribution (r.c.d.) of $\left(X_{\tau+t}\right)_{t \in I}$ given the sub-σ-algebra $ \mathcal F_τ$ (this notation is defined at the end of the definition of a Markov process).
The question is: why does the remark emphasise that the index set $I$ be countable? I understand that this would guarantee existence of the r.c.d. (since then $E^I$ is polish, so $(X_{\tau+t})_{t \in I}$ takes values in a polish space and thus has a r.c.d. given $\mathcal F_τ$), but I don't see why we need this guarantee: It seems to me that if the strong Markov property holds, then $(ω,A) ↦ κ(X_τ(ω),A)$ is a stochastic kernel which by $(*)$ is a r.c.d. of $(X_{\tau+t})_{t \in I}$ given $\mathcal F_τ$. On the other hand, if $(\diamond)$ holds for every a.s. finite stopping time $τ$, then $(*)$ follows from properties of regular conditional distributions (specifically Theorem 8.38 in the same book).