Good subject and it just so happens I've studied and photographed THer holes here in Michigan for many years.
There are different sod situations at different locations and times of the year. In some areas where the ground is kept wet via underground sprinkling systems, etc., the flap method and placing the dirt on a piece of plastic can work but the dig is still visible to some extent.
The best time to dig holes and show the least damage is early in the spring while the grass is still dormant. Usually by summer the damage will fade if the hole was made very carefully and the sod receives enough moisture.
On dry sandy ground where the grass struggles for life and depends on lean rains it's impossible to dig any kind of hole without it showing a few days later (see attached picture). The brown spot in this photo will be visible until next summer and possibly longer. Here in Michigan I've observed carefully dug 3-4 inch metal detecting holes that were clearly visible for up to 3 years.
I know many don't agree with this but I've been at this game a long time and I've tried every recovery method there is. Some methods produce less damage, but all methods make some damage.
Personally, I only hunt parks that are very poorly kept and I hunt private yards only in early spring.