Supercell band structure unfolding advantages

Some reasons why you may want to do band unfolding have been explained by Jack. What I would like to add here concerns your second question about doing band structure calculations in the supercell: you don't want to do that.

Supercells are typically needed when studying non-periodic systems using calculations with periodic boundary conditions. These could include the study of point defects (non-periodic in all 3 dimensions), line defects (non-periodic in 2 directions), surfaces or interfaces (non-periodic in 1 direction), etc. In all these cases, you build a supercell that is long along the non-periodic direction (or directions) so that you attempt to approach the non-periodic limit. In reality, what you have is still a periodic system in that direction, but the period is large enough that it is indistinguishable from a truly non-periodic system (of course this is the ideal, in practice you may not be able to use a large-enough supercell to reach this desired limit).

So what does this all mean for a band dispersion? A band dispersion is a relation between the energy and momentum of an electron, $E(mathbf{k})$, where the momentum $mathbf{k}$ takes some allowed value in the first Brillouin zone. We need to distinguish two scenarios:

1. Real non-periodic system. In a real non-periodic system, the size of the Brillouin zone along the non-periodic direction(s) is zero, that is, there is no Brillouin zone along that direction. This means it makes no sense to talk about a band dispersion along the non-periodic direction. The correct way to think about this is in terms of densities of states, which are well-defined even for a non-periodic system.
2. Simulating a non-periodic system with a supercell. In this case, you will have a very long supercell along the non-periodic direction, which means that you will have a very short Brillouin zone along that direction, but crucially it will not be zero. So in principle you could plot/calculate a dispersion along this short Brillouin zone direction as you suggest. However, this dispersion has no physical meaning. You will get band folding, so a large number of overlaping bands. The larger the supercell, the more bands you will get, until they form a sort of continuum. This is actually the band folding slowly building up the density of states of the truly non-periodic system, which is what is really meaningful in a non-periodic system. Therefore, my advice would be to focus on the physically meaningful density of states when attempting to simulate non-periodic systems.