This is a subject I've wondered about for some time. How many players could the Celtics draft and "stash", ie won't be on the 2016-17 team? I assume all five 2nd-round picks, but I'm not sure.
The team could absolutely roll with 3 more rookies (first rounders) next year, given that Sullinger, Turner, and Zeller are all free agents and Amir Johnson and Jerebko's contracts will be non-guaranteed. The question is, as you've mentioned already, Do you really want to replace three or more veterans with three rookies?
If you add three rookies, you'll have 6 players with 0-1 year of experience on the team, plus James Young. That's 7 guys with little or no time on the court or proof of being able to stick in the league.
This is clearly not ideal. But I think it's important for Ainge not to be so afraid of this scenario that it paints him into a corner in the offseason. If there are no good opportunities to trade youth for established talent, or to consolidate draft picks, then you roll the dice three more times and see what sticks. You just have to plan on some of those younger guys actually getting minutes next year, because you can't have all of them riding the bench the whole year.