I see no point in putting a bird in a smaller cage until tamed unless you plan on forcing the bird to like you. I prefer to earn the birds trust and for that the cage size does not matter. A bird also does not need to be removed from their cage to start working with them and earning their trust, or to start training them.
The only time I put a bird in a smaller cage is if the big one freaked them out. Sometimes a smaller cage feels more secure at first. It has nothing to do with training though and mine ended up in the big cage after a week or so.