It turns out solutions to Ito diffusions with Lipschitz coefficients are Feller processes; you can find this in, say, Oksendal's book (on SDE). Moreover, Feller semigroups (semigroups associated to Feller processes) are strongly continuous and even contracting on $C^0$ under the $C^0$ topology, and this is a subtle point. We'll explain this argument and your concern with $L^2(\mathbb{R})$ below.

Let's look at $f \in C^{\infty}_0(\mathbb{R})$ first, and we'll also assume moment bounds. By Taylor's approximation, we'll assume we can write $$f(X_t) \ = \ f(X_0) \ + \ f'(X_0)dX_t \ + \ \frac{f''(X_0)}{2} dX_t^2 \ + \ \ldots$$

Taking expectations (and writing $dX_t$ in terms of its Ito diffusion representation), we see $$P_tf(X_0) \ = \ f(X_0) \ + \ f'(X_0) \mathbb{E} b(X_t) dt \ + \ \frac{f''(X_0)}{2} \mathbb{E} |\sigma(X_t)|^2 dt \ + \ \ldots$$

This is the standard calculation for computing generators of diffusion processes; in particular, we might as well forget the lower order terms above. If we were only concerned about pointwise limits, we'd be done immediately (since the coefficients of the Ito diffusion $dX_t$ are more than nice enough). But we need $L^2$-convergence.

Because of the Lipschitz assumption on the coefficients, we can bound the expectations on the RHS above in absolute value by expectations of $|X_t - X_0|, |X_t - X_0|^2$ respectively. In particular, finding moment estimates for $dX_t$ (which are bounded and continuous in $t$, thinking about quadratic variation of the process $X_t$), the expectation terms on the RHS above are bounded in polynomials in $X_0$. In particular, for each fixed test function $f$, we have the bound $$P_tf(X_0) \ - \ f(X_0) \ \leqslant \ C(|f'(X_0)X_0| \ + \ |f''(X_0)| (|X_0|^2 + |X_0|),$$

I might actually be missing some terms if I computed incorrectly, but this isn't too important. The important idea is that the RHS of the above inequality is in $L^2(\mathbb{R})$ because $f$ and all its derivatives are. Thus, dominated convergence tells us $$\lim_{t \searrow 0} \ \| P_tf - f \|_{L^2(\mathbb{R})} \ = \ \| \lim_{t \searrow 0} \ P_tf - f \|_{L^2(\mathbb{R})} \ = \ 0.$$

To address your question with all functions in $L^2(\mathbb{R})$, note that the semigroup actually acts on $L^2(\mathbb{R})$ and let $f \in L^2(\mathbb{R})$ be any $L^2$-function (since it's Feller and extends by continuity and density). Moreover, suppose $\varphi \in C_0^{\infty}(\mathbb{R})$ approximates $f$ within $\varepsilon$ in the $L^2$-sense. Now consider (I'll suppress all norms explicitly assuming all unspecified norms are in the sense of $L^2$) $$\| P_tf - f \| \ \leqslant \ \|P_tf - P_t\varphi\| \ + \ \|P_t \varphi - \varphi \| + \|\varphi - f\|.$$

We can choose $t$ small enough so that the last two terms on the RHS are bounded by $\varepsilon$. The problem is the first term, which is where the condition you gave would be useful. Note $P_t$ evaluated at an arbitrary point $x$ defines a (positive) continuous distribution on $L^2(\mathbb{R})$ (since it does so on the dense subspace). Thus, choosing $\varphi$ close enough to $f$ in the $L^2$ sense, note first we can assume $f$ is supported on the support of $\varphi$ losing an $\varepsilon$ term (by dominating the tail of $f$ by a tiny Schwartz function), and we know $|P_t\varphi - P_t f|_{L^{\infty}}$ is bounded by $\varepsilon$. Since $\varphi$ has compact support, everything ends up bounded by arbitrarily small $\varepsilon$, which tells you that such diffusions have strongly continuous semigroups on $L^2(\mathbb{R})$.

Just a quick remark: please let me know if anything is not clear or seems false. Thanks for the problem!