= Solution
For $\kappa>0$, divide the <Boundary-point Bessel flow for SLE> by $\sqrt\kappa$ and replace $W$ by $-W$. The resulting <Bessel process> has dimension
$$
\delta=1+\frac4\kappa,
$$
since its drift is $\tfrac{\delta-1}{2R_t}=2/(\kappa R_t)$. The <Hitting-zero classification for a Bessel process> suggests the threshold $\delta=2$, or $\kappa=4$. Here is a direct verification including accessibility in finite time.
The <infinitesimal generator> of $X$ is $\mathcal L=\tfrac\kappa2\partial_x^2+(2/x)\partial_x$. An increasing <scale function of a one-dimensional diffusion> is
$$
s(x)=\begin{cases}
\dfrac{x^{1-4/\kappa}}{1-4/\kappa},&\kappa\ne4,\\
\log x,&\kappa=4.
\end{cases}
$$
It satisfies $\mathcal Ls=0$. For $0<\varepsilon<b<R$, the <boundary hitting probability from a diffusion scale function>, or <optional stopping theorem> applied to $s(X)$, gives
$$
\mathbb P_b(T_\varepsilon<T_R)
=\frac{s(R)-s(b)}{s(R)-s(\varepsilon)}.
$$
For $0<\kappa<4$, $s(\varepsilon)\to-\infty$, so this probability tends to zero as $\varepsilon\downarrow0$. A finite zero hit has a bounded path before the hit and hence precedes $T_R$ for some integer $R$; taking a countable union proves that zero is never hit.
For $\kappa>4$, write $a=1-4/\kappa>0$. Taking the inner boundary to zero gives $1-(b/R)^a$. This limit really concerns a finite zero hit: the <Itô formula> gives
$$
d(X_t^2)=(\kappa+4)dt-2\sqrt\kappa X_t\,dW_t.
$$
Stopped on $[\varepsilon,R]$, it implies
$$
\mathbb E_b(T_\varepsilon\wedge T_R)
\leq\frac{R^2-b^2}{\kappa+4},
$$
uniformly in $\varepsilon$. The increasing limit of these exit times is finite <almost surely>; the stopped squared process has a continuous extension, so its limiting lower endpoint is zero. Thus
$$
\mathbb P_b(T_0<T_R)=1-(b/R)^a.
$$
Let $R\to\infty$ to obtain $\mathbb P_b(T_0<\infty)=1$. When $\kappa=0$, the explicit solution is $X_t=\sqrt{b^2+4t}$, which stays positive.
Combining this with the <SLE boundary swallowing criterion> gives \b[the critical parameter and the two regimes]:
$$
\boxed{\kappa_c=4,\qquad
\mathbb P(\gamma\text{ hits }[b,\infty))=
\begin{cases}0,&0\leq\kappa<4,\\1,&\kappa>4.\end{cases}}
$$
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