The Eagle-Eyed Pixel Peeper and the Perfunctory Print Peruser might well disagree on whether the DoF in a given image is exactly 1 meter or 2 meters. However, they can't help but agree on the overall relative sharpness effects of aperture choice. For a given sensor, focal length, cropping, and subject matter, changing the aperture of the lens changes the sharpness of the scene is nearly universal ways:
At wide-open aperture:
- the in-focus center of the image is OK but could be sharper
- most lenses show non-uniform sharpness across the frame -- the edges and corners are much softer than the center.
- the range of subject distances that are "in focus" is quite narrow.
Stopping down a bit from wide-open toward the sweet spot:
- the in-focus center of the image becomes sharper than it was wide-open
- most lenses show increasingly uniform sharpness at the in-focus distance across the frame.
- the range of subject distances that are close to the in-focus sharpness grows larger.
Stopping down more than the sweet spot:
- the in-focus center of the image becomes softer due to diffraction
- most lenses show very-uniform sharpness across the frame.
- the range of subject distances that are as sharp as the center-in-focus spot is even larger.
The point is that:
- wide open: only a narrow back-to-front slice of the center of the frame is in-focus
- sweet-spot: the center is as sharp as the lens can get, edge-to-edge is decently sharp, and some range of front-to-back is sharp
- closed-down: much of the front-to-back and corner-to-corner of the scene has uniform sharpness but that sharpness is not as good as the sweet spot sharpness
Thus, aperture choice is important if the photographer wants to either isolate the subject, have edge-to-edge uniformity of focus, or ensure that everything from some close foreground to a deep background is in focus.
As for using a DoF calculator, the calculators do work but each photographer needs to figure out their personal tolerance for the circle of confusion (CoC). That depends on the photographer's chosen format, their personal assessment methods (e.g., a print of some size vs. pixel peeping) and their personal subjective sensitivity to OOF. Once they know their CoC, they can calculate DoFs and hyperfocal settings that work for them.
(Note: "most lenses" means most lenses that are not decentered and that have decently flat fields. Decentering and field curvature can harm edge and corner sharpness even when stopped-down.)