Why isn't the original force transmitted to m2
In order for the two masses to accelerate together due to the pulling force $F$ on $m_1$, the pulling force on $m_2$ must be less than $F$, regardless of the magnitudes of the two masses.
To see this, refer to the free body diagrams (FBD) below, which assumes the only net external force on the system of masses is $F$, and that the strings are massless and inextensible.
From the FBD on $m_1$, for $m_1$ to accelerate to the right the tension force $T$ to the left must be less than $F$. From the FBD on $m_2$, the only external force acting on $m_2$ is the tension force $T$ to the right. Therefore the force acting on $m_2$ must be less than $F$.
Hope this helps.
