The problem is that minor factions are designed so that the power level of their cards is slightly higher than the power level of the major faction, and this is balanced by the fact that you can only have 12 of them in a deck. If you could make an entire deck out of minor factions, it would be terribly hard to balance.
I'm leaving aside the fact that you would have to add faction HQ for them, or at least some symbolic Elite cards xD Too much work and too much chaos would arise in terms of balance.