I know the usual method of showing that an Erdos-Renyi random graph $G(n,p)$ is connected is by letting $X_k$ be the random variable counting the number of connected components on $k$ vertices and then showing $\Bbb E(\sum_{k=2}^{n/2}X_k)\to 0$ and then finding when isolated vertices vanish.
I also know that a branching process is used to explore the connected component around any vertex $v$ to find the size of the various connected components.
My question is: could this second method be utilised to also find the threshold where a random graph is connected? If the branching process can compute the expected size of components, if we are able to show there are no components of size $1\leq k \leq n/2$ we would be done right? Is this an approach that has ever been considered before and if so is there any literature on this?