Finite coupling makes connectivity probability monotone

Give each possible edge a coordinate with three outcomes $0,1,2$ of respective weights $1-q,q-p,p$, and use their finite product. These weights are nonnegative and sum to one. Include an edge in $H_p$ if its coordinate is $2$, and in $H_q$ if its coordinate is $1$ or $2$. Summing coordinate weights shows the marginals are respectively $p$ and $q$; summing product weights over each prescribed coordinate block factors, so the graph marginals are $G(n,p)$ and $G(n,q)$.

On every outcome the two graphs have the same nonempty vertex set $[n]$, and every edge of $H_p$ is an edge of $H_q$. Each path of $H_p$ therefore remains a path of $H_q$, so the event $A$ that $H_p$ is connected is contained in the event $B$ that $H_q$ is connected. For the product outcome weights $w$, nonnegativity gives $\mathbb{P}(B)-\mathbb{P}(A)={\sum }_{\omega \in B\setminus A}w(\omega )\ge 0$. By the graph marginals in step 1.1 this is the desired inequality, including all endpoint cases.