We've done HB and AI and I don't think that is the issue regarding freedom, it is more about the hotel location, surrounding facilities and cost of using those facilities. The AIs we've had except the last were close enough to restaurants/pubs for members of the party to eat out whenever. The last one in the DR was not, but there was no real need due to the hotel facilities and quality.
What extra freedom do you get with HB, you still have paid for breakfast and evening meal, but they are fixed times, so you either fit in or pay extra to eat out same as you would with AI.
With the HB we found it annoying to say the least that we had to keep forking out during meal times for drinks, even water.
At our last AI, they started breakfast at 7am, from there on it was possible to get something relevant to eat or drink somewhere in the hotel until 6am, so you actually had the freedom to eat if and when you wanted to.
Again it comes down to the definition of freedom. The main plus with AI is the drinks for kids whenever they want them, and the freedom you get by not having to carry money etc. around with you while you are in the hotel. We spent 2 weeks in basically cossie, towel and wristband. This freedom from worrying about where to put the valuables if you want to go in the water, have we got enough for the drinks this morning, woops forgot the icecream costs etc is what I enjoy most. We certainly don't tie ourselves to the hotel though.