The reason that TT (and especially triathlon) bikes have the butt forward is so that being low and aero doesn't require a super closed hip angle. This ends up putting a lot of weight on the upper body.... but aerobars put vertical support near the elbow, which helps make that easier, and simultaneously gives you nice aero level forearms.
If I'm trying to do a fast couple-minute effort on my drop bar bikes on level ground, I'll roll my hips forward until a narrow part of my sit bones are resting way out on the nose of the saddle, put my hands in the hooks, and hold my forearms level. I'm not just leaning my torso forward, I'm rotating my entire body forward relative to the bottom bracket; my torso-leg-pedal relationship barely scrunches up at all, and I can still breath and pedal nearly as well as I can if I'm riding in more "normal" positions... albeit with extra upper-body effort.
This is more or less what the old phrase "on the rivet" refers to. Although ironically, my only saddle with a rivet out front - a Brooks Swift - has a nose that's too narrow for me to roll forward onto.