Although the image our eye forms on the retina has depth of field, our brains suppress the out of focus areas and it is very difficult to see that parts of the world at different distances are not all in focus. Because accommodation is so fast, we are unaware of shifting from near to far objects and we see both near and far objects as in focus. This is why paintings never had depth of field before the appearance of photography and rarely (if ever) after. So you are in good company in not using focus as a marker of visual importance.
Even within the walled garden of photography depth of field was never as big an issue with large format cameras, which are not restricted to a film plane perpendicular to the lens axis so depth of field is not determined by aperture, focal length, and focus distance. With film cameras, print resolution is determined by the size of the negative, because there is a fixed relationship between the negative and the print. That is not the the case with digital cameras. A 100 MP Hasselblad sensor has 4 micron photosites. The resolution and dynamic range are exactly the same as every other sensor with 4 micron photosites, regardless of sensor size. There are more photosites, but that just means you can print larger, not, as with a film negative, that you can have more resolution in the same print size.
For me it is easy to see that when I am looking at an object such as the computer monitor that I'm using to post this answer, that the areas of my visual field which are substantially at a different distance from the monitor are blurry (even if they are in the foveal area. However, the magnitude of the out of focus effect is smaller than what one can achieve with a large-aperture lens on a typical large sensor camera. A more obvious effect in human vision is that what we are looking at in the center of the visual field is sharp and surrounding areas are blurry, which is separate from the effect of depth of field.
Paintings were made typically sharp across the field not because human vision presents them like that but because this allows us to look at the painting like we were looking at the scene, focusing our attention on different parts of the image with the sharp central vision which then shows the details of that part of the scene / painting. The painting shows the scene open to exploration by the viewer. If they blurred the background or foreground, it would lock the viewer into looking at the main subject probably too strongly. Since the painter controls every aspect of the scene as represented in the painting, there is no need to blur out unwanted clutter as we often do in photography with a large-aperture or telephoto lens. A photographer (at least traditionally) does not have this control (except in post-processing) and thus photographers often choose to use a large-aperture lens to make the image cleaner and with fewer uncontrolled details as it can be hard enough to get the picture taken at the exact moment when the subject shows a particular expression or the geometry is pleasing. In studio photography, by contrast, usually we have complete control over what is shown in the photograph, including the lighting and background, props, and so often deep depth of field is used (I usually shoot at f/8 to f/13 in the studio). However, I don't have such control outside of the studio and I don't want to change the environment explicitly, so I often use a wide aperture to limit the degree of chaos shown in the surroundings. I prefer not to remove or blur objects in the photograph in post-processing as this to me goes against the spirit of photography and I still consider it an act of documentation to a point. Since I'm so used to large aperture photography, if I do need to show everything in focus and if it includes people, I find myself in great difficulty choosing the right moment where everything is in its place (without controlling the subjects completely, which I do not want to do). It's a paradoxical enterprise. ;-) I don't want to exert control, but use the tools of photography to select something from reality that leads to a pleasing or informative photograph in my way of thinking.
By the way Leonardo did not think objects should be drawn with precise boundaries at least according to a documentary I was watching a few days ago. So the objects were a bit blurred and sharp outlines were avoided in his paintings. I think the results look natural but not photorealistic obviously. The level of detail possible with modern photography was not realistic to paint by hand, I'm not sure if he would have wanted to do that even if he could have. It would seem he went for painting with techniques that made the subject feel natural and three-dimensional but not with the level of detail that photography (which had not been invented) gives. Of course, a lot of photographers also blurred (and blur) their portraits either using a soft-focus filter or lens, a process which didn't lead to a great deal of detail, or in post-processing. But this is a separate discussion from depth of field.