Let $f_0(S_T) =f(S_T|S_0)$ be the risk-neutral PDF for the underlying asset price at time $T$ (conditional on the price $S_0$ at present time $t=0$). The probability that the price is above a strike price $K$ at time $T$ is
$$P(S_T \geqslant K) = \int_K^\infty f_0(x) \, dx.$$
This is just definitional regardless of the shape of the distribution (eg. symmetric, skewed, etc). It could be the distribution implied in the known option prices at $t= 0$.
The price of a vanilla call option at $t=0$, expiring at time $T$ and with strike price $K$, is the discounted risk-neutral expected value
$$C(K) = e^{-rT} \int_0^\infty\max(x-K,0) \, f_0(x) \, dx = e^{-rT} \int_K^\infty(x-K) \, f_0(x) \, dx. $$
Here we have ignored dividends and suppressed the dependence of the option price on other parameters in writing $C(K)$.
We can apply the Leibniz rule and differentiate the integral once with respect to $K$ to obtain
$$\frac{\partial C}{\partial K} = -e^{-rT}\int_K^\infty f_0(x) \, dx \\ \implies P(S_T \geqslant K) = - e^{-rT}\frac{\partial C}{\partial K}$$
In the presence of an implied-volatility skew, the underlying distribution is not lognormal. However, we can represent the option price as a composition of the Black-Scholes formula with a (smooth) implied volatility as a function of strike:
$$C(K) = C_{BS}(K,\sigma(K)).$$
Hence,
$$P(S_T \geqslant K) = -e^{-rT}\frac{\partial C_{BS}}{\partial K} - e^{-rT}\frac{\partial C_{BS}}{\partial \sigma}\sigma'(K) \tag{*} $$
Typically for an equity index, the skew exhibits a negative slope, $\sigma'(K) < 0$, and vega, the partial derivative of $C_{BS}$ with respect to $\sigma$, is positive.
All else the same, $P(S_T \geqslant K)$ increases as $-\sigma'(K)$
increases -- ie., the skew steepens.
As you observed, the binary or digital call option $C_D$ can be replicated approximately with a call spread according to
$$C_{D }(K) \approx \frac{C(K-\delta) - C(K+\delta)}{2\delta}.$$
As the strike spread $2\delta$ tends to $0$ and the notional $1/(2\delta) $ tends to infinity, the replication is more accurate (although impractical) and
$$C_D(K) = \lim_{\delta \to 0} \frac{C(K-\delta) - C(K+\delta)}{2\delta} = - \frac{\partial C}{\partial K}.$$
This, of course, shows that the digital option price itself is directly related to the probability of the underlying ending above the strike at expiry.
