Feller condition guarantees the right continuity of the joint measure. In your argument $\begin{align*} \lim_{t \to 0}P_{t}f(x)\overset{\Delta}{=}\lim_{t \to 0}E_{x}[f(X_t)]=E_{x}[f(X_0)]=E_{x}[f(x)]=f(x) \end{align*}$
The second equality will fail without Feller property because
$$\lim_{t \to 0}E_{x}[f(X_t)]=lim_{t\rightarrow 0}\int_\Omega f(X_t(\omega))dP_x(\omega) = \int_\Omega lim_{t\rightarrow 0}f(X_t(\omega))dP_x(\omega) \overset{?}{=} \int_\Omega f(lim_{t\rightarrow 0} X_t(\omega))dP_x(\omega)=E_{x}[f(X_0)]$$ even with dominance on integrand sequence $\{f\circ X_t\}$, you can only proceed the second equality but not necessarily the third one(You cannot argue by continuity of $f$ because $X_t(\omega)$ is not only one sequence), which is guaranteed by Feller.