As explained in my older answer Joe linked in a comment, the purpose of stretch and fold is to align the gluten sheets, producing the typical structure of kneaded bread. Depending on your final shaping, you end up with either a sheetlike structure (e.g. in ciabatta) or with spirals/threads in kozunak and other braided breads.
One reason to not do the stretch and fold is when you want to achieve a cakelike crumb with yeast dough. The mixer does develop the gluten, but it makes a regular crumb with no discernible direction of the dough structure. For Pullman style loaves, or also for some types of "cake" made with yeast dough (sorry, English doesn't have the correct word here) such as zwetschgendatschi, a mixer will produce better results. From your list, the hamburger buns will be closer to fast food restaurant style if you use the mixer, it's up to you to know if you want them this way.
The other, and much more common reason, is convenience. While kneading dough can be a very pleasant, relaxing thing to do, sometimes you just value your time over the perfect bread texture, and do it in the stand mixer, so you can take care of other stuff in the meantime.
The third reason why someone would choose the mixer is to avoid the learning curve of doing stretch and fold, or not knowing that the results of proper hand kneading technique are superior. This is, in my eyes, a false economy, since learning to knead properly is easy and a skill worth having even for casual bakers.