Best pot size varies on growing medium and technique. Anything larger than 3 gallons, perhaps less than 3, is a waste of space with coco and autopots. I'm guessing, but I suspect that plants would get just as big in a couple gallons with coco and autopots.
OTOH, if you want to do a soil grow with only water, three gallons will not do the job. Neither will 5 gallons or maybe even 7. If I try a soil and water grow this winter, I will use at least 10 gallon pots, maybe larger, and I will still plan on supplementing if the plants ask for it. Totally different picture than coco.
As to the original question about transplanting, one general reason to consider it is that a smaller initial pot can, according to general horticultural experience, generate better root development than starting without constraints of a small initial pot. Dunno how much this applies to autos, but transplanting carefully has never hurt my plants yet, so I will likely continue to do it.
Another reason that matters to me is that starting in small pots allows you to do more initial plants in a smaller space in which temperature and humidity might be easier to control while the plants are young, and the larger number allows weeding out of weaker plants before committing the final pots which you have limited space for.
The same logic applies to germinating seeds. I like to start with an initial soak to allow healthy tap roots to show up. Waiting for a week or more for an untested seed to emerge from a final pot only to find that it didn't show up after all doesn't work for me. At least if I see a nice tap root, I know that the seed is likely worth planting.
Happy growing
@Gary Laser Eyes