The measure should give large values if knowing the value of $𝑥_i$ for a given data point allows one to predict the class with certainty. Conversely, it should give low values of the feature provides no information on the class.
This sounds like a few related things would work for you:$
\DeclareMathOperator{\E}{\mathbb{E}}
\DeclareMathOperator{\HH}{H}
\DeclareMathOperator{\MI}{MI}
\newcommand{\ud}{\mathrm{d}}
$
Mutual information
The mutual information between $X_i$ and the label $Y$:
$$\MI(X_i, Y) = \HH[Y] - \HH[Y \mid X_i]
= h(\Pr(Y = 1)) - \E_{X_i}[ h(\Pr(Y = 1 \mid X_i) ],$$
where $h(p) = - p \log p - (1-p) \log(1-p)$ is the entropy of a Bernoulli distribution. If you train a probabilistic classifier for $Y$ given $X_i$, this gives you a reasonable score. There are also various more direct mutual information estimators, or entropy estimators, but since $Y$ is so simple I think the classification approach makes sense. Incidentally, this is closely related to the Jensen-Shannon distance that you suggested.
Distances between distributions
You can also consider a direct notion of distance between the distributions $X_i \mid Y = 0$ and $X_i \mid Y = 1$.
$f$-divergences
Something like the Jensen-Shannon as you mentioned, or the total variation, would be fine (assuming everything has a density, or at least one divergence is absolutely continuous wrt the other). These are $f$-divergences.
The distances themselves correspond roughly to how directly distinguishable the distributions are by a perfect classifier: if $Y$ is exactly the seventeenth-order bit of $X_i$, no ML method that looks at the $X_i$ as a continuous value is ever going to recognize that, but the total variation will still be maximal.
So, estimating them can be arbitrarily hard if you don't assume anything about the distributions. But if everything has relatively smooth densities and you have a reasonable number of samples, since you're in one dimension there should be no serious challenges. Options include, as you mentioned, a plug-in estimator based on the KDE or similar, corrections to that estimator, estimators based on $k$-NN density estimators, estimators based on convex optimization (related to training classifiers), and many more.
Integral probability metrics
There are also other options for distance metrics, where the metrics themselves are more constrained and hence easier to estimate. One nice class is called integral probability metrics: $$\mathcal D_{\mathcal F}(P, Q) = \sup_{f \in \mathcal F} \E_{X \sim P} f(X) - \E_{Y \sim Q} f(Y).$$
These are defined by the choice of test functions $\mathcal F$. If you pick $\mathcal F$ to be the set of functions with outputs bounded in $[-1, 1]$, you get the total variation.
In 1d, one particularly nice option is the Wasserstein distance (aka earth mover's, Kantorovich-Rubenstein, Mallows, ...); it has a simple closed-form estimator in 1d, and corresponds to how well your distributions can be distinguished with Lipschitz functions.
Another option that's particularly nice in higher dimensions, though it also works well here, is the maximum mean discrepancy, based on the kernel trick. Statisticians might be more familiar with the energy distance, which is a special case of the MMD. Its square has a simple unbiased, asymptotically normal estimator. For many choices of kernel, the distance will be zero if and only if the distributions are the same. But its notion of "more similar" is fundamentally connected to the choice of kernel, in a way that is also of course analogously true for any estimator you might use of any distance, but is more explicit here.