First, everyone's assuming the last bike had the best position.
It seems to me if you bring the saddle forward if you want to keep the hip angle the same you don't want to just push the bars out. Rather you want to rotate your hips and bars about the bottom bracket axis. That means raising the saddle slightly to preserve saddle-to-BB, moving your bars out, but then dropping them slightly.I did some calculations here
Basically you want to povot this diagram about the bottom bracket:
The net result:
ΔyS = ‒Δx (xS / yS)
ΔyH = ‒Δx xH / yS.
ΔxH = Δx yH / yS,
Check the link for details.
But this is just simple geometry. Honestly I know nothing about proper fitting.
Yes, you are right - if you bring the saddle forward and maintain the same saddle height measured to the BB, you need to raise the saddle a little, so you are indeed effectively rotating around the BB.
My last bike fit moved me forward a bit amongst other things, and it seems to work quite well for me. As mentioned above, it changes the angle the hips make with the torso when pedalling. The fitter seemed to think this would give me more power (it certainly hasn't reduced it), but it's also commonly said that sitting further back can give more power (especially for climbing) by activating the glutes more.
I just sit a bit further back on the rear edge of the saddle on the rare occasions I feel I need to, but you will often see people do the opposite, i.e. have the saddle set further back and then move forward to be "on the rivet" when they want more power on the flat. Me, I find it more comfortable very occasionally sitting further back on the saddle than frequently sitting on the nose without my sitbones supported...