It would be quicker using it, for the fairly simple reason that you'd get more control over the puppets, and not have them wasting builds on barracks, armories, etc. Therefore the puppets would produce more culture, gold, science for you, without increasing the cost of anything, as you'd raze them before taking your final group of policies. In my most recent game, that would have been 1750 if I didn't do a UN experiment and win via diplo, I had 8-10 puppet cities when I started saving up my culture for Cristo Redentor. Full control over those would have meant less money wasted, more culture buildings, etc. Would just have to make sure national wonders like hermitage were built before annexing. It would also mean you could found more cities yourself if needed.