The main effect of changing the overhang is that it changes the geometry of the initial loop formation - the longer the overhang the tighter the loop. As a thought-experiment imagine a very long overhang - in such a case at the end of the power stroke (assuming for the thought experiment purposes that energy was actually transferred to the line) the fly line basically keeps going in a straight line until after the release of the shooting line dragging the shooting line behind it rather than turning over. This would then be in effect a “zero” inch loop but with no turnover!
Interpolate between zero and infinity overhang and you get the idea. The tighter loop produced in this way can help your casting a lot, but as implied by the original question the timing becomes more and more critical as the overhang gets longer. So YES, when casting a dead chicken or anything you feel like is putting more pressure on you to get good energy transfers a lot of people will both use a little less overhang and do the usual, which is to “round” their stop a bit to intentionally get a wider loop, and a more stable and reliable energy transfer to compensate for the heavier load. In that case “style-be-dammed, we need to get this puppy airborne”.
Same goes when the “load” is due to a extra-large or extra-sticky amount of sunk line at the moment of the power stroke.
BTW a very extreme rounding to the extent of virtual complete elimination of the stop altogether is a great tool (even though you may look like a “tool” to people watching and unaware of what you are doing) for casting safely under overhanging branches on the opposite bank. Learning that trick can be a real lifesaver if you face that sort of thing a lot, and you would in addition of course use zero overhang in those case.
You can also adjust the overhang, as mentioned, simply to change/recalibrate amount of line you are casting relative the water when you change your wading depth. But IMHO once you get used to adjusting on the fly (no pun intended) to the difference between skagit length, scandi length, and short belly length lines you will most likely never even think about that part. You just watch the loop and adjust your cast accordingly unless things are so extreme as to be a “trick cast”, like choking up to flick the cast under overhangs on your side of the river. Personally while I know I’m changing my cast mechanics to compensate for the wading depth I feel like it is almost subconscious at this point.