Dimensional analysis is enough to see that $mc^2$ has the same units as energy while $\gamma mc$ or $mc$ doesn't.
Concerning the components of a 4-vector, special relativity unifies the spatial and temporal components. But the 4 components of a 4-vector with "uniform units" do not necessarily enjoy the same normalization as the quantities outside relativity. Instead, you must typically multiply the time component by $c$ or $1/c$ to get the usual non-relativistic normalization of the quantity.
The position vector is $(x,y,z,ct)$. Note that all of them have the dimension of length. But the real time is $ct$, the last component, divided by $c$. Similarly, the energy-momentum vector has $m_0\gamma v_x,m_0\gamma v_y, m_0\gamma v_z,m_0\gamma c$. Again, all components have the same units but now you have to multiply the last component $m_0\gamma c$ by $c$ to get the usual normalization for the energy, $E=m_0 \gamma c^2$.
This is just a question of units. Nothing guarantees that the simplest identifications and conventions will lead to correct formulae without any extra $c$ with the units used before relativity. When we say "the time component of a vector is a particular quantity known before relativity", we mean that it contains the same information but sometimes we need to normalize it differently, add a universal factor. Before relativity, people used very unnatural units for many quantities and the temporal and spatial components didn't have the sam units even though their key information was always a part of the same 4-vector which shows that they may be "rotated into each other".
Among physicists who study relativistic phenomena (e.g. particle physicists), this is a complete non-problem and physicists often use units with $c=1$, anyway. This really means that distances are measured in light seconds and the speed of light is one light second per second, and the difference between a light second and second is suppressed. (Particle physicists typically use units in which one GeV, and not one second, and its powers are used for everything, i.e. times, distances, energies, momenta, masses, etc.: they also set $\hbar=1$ so that energy and frequency i.e. inverse time have the same units.)