Statement: if $X$ is a finite path connected CW-complex, then $X$ is homotopy equivalent to a CW-complex with only one $0$-cell.
Thoughts: we can use the theorem that for a contractable subspace $A \subseteq X$ such that $(X, A)$ has the homotopy extension property, the projection $X \to X/A$ is a homotopy equivalence. I want to apply this for each $1$-cell that does not have its ends glued together. This is ok because then it is a contractable, there are only finitely many, so eventually this procedure will stop and $(X, A)$ is a relative CW-complex, so it has HEP.
We will be left with only $1$-cells that have their ends glued together. I'd like to conclude from path connectedness that there is only one $0$-cell left, but could not manage this.