It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is asymptotically almost surely connected. The way I know how to prove this is (1) first counting the expected number of components of order $2, 3, \dots, \lfloor n/2 \rfloor$, and seeing that the expected number is tending to zero. Then (2) showing the expected number of isolated vertices is also tending to zero.

This approach also allows more precise results, such as: if $p = (\log{n}+c) / n$ with $c \in \mathbb{R}$ constant, then Pr$[G(n,p)$ is connected] $\to e^{-e^{-c}}$ as $n \to \infty$, which follows once we know that in this regime the number of isolated vertices is approaching a Poisson distribution with mean $e^{-c}$.

I am wondering if it is possible to give an easier proof (of a coarser result) along the following lines. There are $n^{n-2}$ spanning trees on the complete graph, and $G$ is connected if and only if one of these trees appears. So the expected number of spanning trees is $n^{n-2}p^{n-1}$. One might expect that if this function is growing quickly enough, then with high probability $G(n,p)$ is connected.

I think I remember reading somewhere that this approach doesn't quite work --- for example the variance is too large to apply Chebyshev’s inequality. What I am wondering is if there is some way to fix this if we are willing to make $p$ a little bit bigger. In particular, what about $p = C \log{n} / n$ for some large enough constant $C > 1$, or even $p = n^{-1 + \epsilon}$ for fixed but arbitrarily small $\epsilon >0$?