I think it's more about how and what you play. When I played bari in a saxophone octet and we played songs like "My Funny Valentine" I had to play in a differnt way. Often long legato tones. When I play Rockbari it's more about to play in a percussion-like way. The setup was differnt as well. I played a Brilhart HR in the octet and I use(d) a Dukoff D9 in Rock. Both idioms need air but you must use the airstream in a differnt way.
I also think we can forget the talk about that a bari needs a big person to fill the tube with air. There was a small guy in Joe Liggins Band ("The Honey Dripper" -45, some people says that it was one of first Rocksongs ....) called "Little" Willie Jackson. He played great bari. On the pictures I have his low Bb bari reaches to his knees.
Have played Bari and alto each 50% of the time. You won't be able to hold the bottom end as long without coming up for air more frequently. See how long you can hold low A on the bari as opposed to Bb on the alto.
With mine it seems to be a greater volume of air but at a slower rate, but over all far less effort. if that makes any sense
I have to blow my alto and tenor but I only need to breathe into the bari mouthpiece. If I blow, then nothing or a squeak comes out. People I know who play sax and has tried my stock mouthpiece just get silence/squeak to start off with. Then when they back right off and relax their throat they cant believe how easy it is. I have tried a Yani B901 and that was more like my alto/tenor in that you needed to blow.
I just asked Jane what she thinks and she says "Loads less effort and yes just lightly breathe into it"