Jon, your description re the white dot/black background is how I've practiced and shot for many years. Except...remove the black background and don't look at the bow hand or bow, lol.
What you are describing in your post (minus looking at the bow and hand) is essentially instinctive shooting. The concept entails centering only that 'spot' in your field of vision, much as you would when using a wide-ring peepsight on a rifle, and simply allowing the rest of what you see to blur out...just as with a peepsite, until the spot is centered in your field of view. Everything you see other than the spot essentially becomes a means of establishing both center and distance ...or in other words it's the 'aiming' part of the shot, sort of aiming without aiming. There is no gap interpretation or concious aiming with any part of your equipment, including your bow or hand. Your sight picture becomes the 'aiming' part, by centering the spot in the middle of everything you see and allowing repetition and consistent form to do the rest. Good form, shoot shots and the subconcious remembers.
The method works quite well at closer ranges but as you might suspect it begins to falter as range increases, due to the eye's dependence on depth perception as a means of centering, It's a feel thing, much as it is when you shoot a basketball or throw a baseball from longer vs. shorter distances. The farther away you are the harder it is to feel the correct lob. You might consider it as a furthering of your athletic endeavors such as those you may have enjoyed when playing ball as a youngster, as opposed to a number crunching excercise that you might have endured during math class (today they call that target shooting, lol).
So you really don't need to look at anything except the spot for this approach to work, as long as you shoot at reasonable distances and your form is the same on every shot. AS for those who say instinctive doesn't work or isn't a viable method (especially for hunting), well to that I say poppycock. They either haven't taken the time to learn it properly or don't want to admit their athletic deficiencies, lol.
I will say this, whether using a sight or shooting instinctive, it's really surprising how much difference a slight variation in basic form can make w/ regard to consistent accuracy. And Rick Welch is right, 'holding' is one way to help fine-tune your centering until the sight picture (the aiming part) builds itself. For some this might take only an instant after hitting anchor, for others (like me...a geezer with bad eyesight) it may take a few moments longer.
But it does work. Hope you enjoy the experience.