We can re-write the problem in terms of $W_t$ alone, or, even better, in terms of the drifted Brownian motion $\tilde W_t = W(t) - M t$, where $M$ is the supremum of $|b(s)|$.

Define $$ B(t) = -1 - \int_0^t b(s) ds - M t ,$$ so that $X(t) = \tilde W(t) - B(t)$ up to time $$ \tau = \inf \{ t > 0 : \tilde W(t) \leqslant B(t) \} . $$ Fix $t_0 > 0$ and define $$ \sigma = \inf \{ t \in (0, t_0] : \tilde W(t) \leqslant B(t_0) \} . $$ Since $B$ is a non-increasing function, we clearly have $\sigma \geqslant \tau$, and hence the measure $$\mu(dx) = \mathbb P(t_0 < \tau, \tilde W(t_0) - B(t_0) \in dx)$$ is dominated by the measure $$\nu(dx) = \mathbb P(t_0 < \sigma, \tilde W(t_0) - B(t_0) \in dx) .$$ The latter is, however, just the distribution at time $t_0$ of the drifted Brownian motion $\tilde W(t) - B(t_0)$, killed upon hitting $0$. As you write in the statement of the problem, this is known to have a density function continuously vanishing at zero, and hence $\mu(dx)$ also has a density function continuously vanishing at zero.

It remains to note that $\mu$ is precisely the distribution of $X(t_0)$, up to an extra atom at $0$.

Remark: A more general approach to the problem, which seems to work also when $b(s)$ is an (adapted) stochastic process rather than a deterministic function, would involve showing first that the distribution of $Y(t)$ — and thus also that of $X(t)$ — has a bounded density function (save for an atom at $0$), and then using Chapman–Kolmogorov equation and a comparison argument similar to the one given above to conclude that the density function of the distribution of $X(t)$ goes to zero at $0$.

We can re-write the problem in terms of $W(t)$ alone, or, even better, in terms of the drifted Brownian motion $\tilde W(t) = W(t) - M t$, where $M$ is the supremum of $|b(s)|$.

Define $$ B(t) = -1 - \int_0^t b(s) ds - M t ,$$ so that $X(t) = \tilde W(t) - B(t)$ up to time $$ \tau = \inf \{ t > 0 : \tilde W(t) \leqslant B(t) \} . $$ Fix $t_0 > 0$ and define $$ \sigma = \inf \{ t \in (0, t_0] : \tilde W(t) \leqslant B(t_0) \} . $$ Since $B$ is a non-increasing function, we clearly have $\sigma \geqslant \tau$, and hence the measure $$\mu(dx) = \mathbb P(t_0 < \tau, \tilde W(t_0) - B(t_0) \in dx)$$ is dominated by the measure $$\nu(dx) = \mathbb P(t_0 < \sigma, \tilde W(t_0) - B(t_0) \in dx) .$$ The latter is, however, just the distribution at time $t_0$ of the drifted Brownian motion $\tilde W(t) - B(t_0)$, killed upon hitting $0$. As you write in the statement of the problem, this is known to have a density function continuously vanishing at zero, and hence $\mu(dx)$ also has a density function continuously vanishing at zero.

It remains to note that $\mu$ is precisely the distribution of $X(t_0)$, up to an extra atom at $0$.

Remark: A more general approach to the problem, which seems to work also when $b(s)$ is an (adapted) stochastic process rather than a deterministic function, would involve showing first that the distribution of $Y(t)$ — and thus also that of $X(t)$ — has a bounded density function (save for an atom at $0$), and then using Chapman–Kolmogorov equation and a comparison argument similar to the one given above to conclude that the density function of the distribution of $X(t)$ goes to zero at $0$.