To address the OP..
While our eyes have a limited depth of field, most people are not aware of it because we focus on whatever we are looking at it.
Part of creating eye-catching photos is presenting something not often seen.
Using a narrow depth of field is an artistic tool that allows one to do just that.
In addition, using narrow dof / large apertures allow the photogrpher freedom from some common problems. Namely distracting backgrounds and blur caused by camera shake.
However narrow dof introduces one shortfall to photos. You can only see one thing. There is no option to scan the frame and see what is going on behind or in front of the subject. Once you have seen the main subject, you may as well turn the page and go on to the next image.
So if you like narrow dof, then by all means use it. But get really good at it, or only use it when you need it.
Don't use it as a fixall.
I think most people aren't consciously aware of their eye's shallow DOF, but only in so much as they don't have the structural framework to consider it. We have the vocabulary and reference points to discuss it. This does not mean, however, that they don't see it. My wife certainly sees it in photos, sometimes says things like "I want a camera that makes my pictures look better" and then shows me images with shallow DOF emphasising the subject and blurring an otherwise disturbing background; and she realizes her own vision is like that. Though she has no language for it, she easily sees the difference between an 8x10 print from a 4x5 and one from a d-slr. These are not things many photographers believe the naive viewer of their works can see- I hold that most viewers do, they just have no ability to provide a reference point and language for what they see.
Also, I disagree with the notion that shallow DOF is a tool the average viewer doesn't see often. Most people watch a couple of movies a week, and they see quite a bit of imagery with shallow DOF, it is a common cinematic tool. Cinema is photography at 24 pictures a second with soundtrack... I don't think that enough photographers consider how cinema shapes their viewers' perception of imagery. Perhaps the generally high production value of cinematic imagery causes a false perception of value even with inappropriate use of shallow DOF.
Along those lines, we are all familiar with the 'miniaturizing' effect of extremely selective focus via tilts. Oddly, this seems to affect every one in roughly the same way- and not only those of us who've done a lot of macro photography and are used to fighting for DOF when we do. But even those who've never even touched a macro lens also seem to associate selective focus with miniaturization. Apparently we are trained by cinema and other media to perceive it that way (as Thebes just noted).
Keith, I've often wondered about that. I grew up with model trains and I saw a lot of photos of actual tiny towns, but I wonder how everyone seems to make that connection. There is even a "diorama" filter on the Olympus EP-2, but I shouldn't say anymore about that here. Maybe it is from the cinema. Interesting question of perception I think.
Erik, when I started my 1:1 stuff, locking the bellows on an LF camera to ensure true 1:1, I was immediately ocnfonted with the fact that only the objects in the plane of focus will be 1:1. Hence we perceive, for good reason, that objects outside the plane of focus (= out of focus elements) are necessarily rendered at some other scale. As for how this works with the eyeball, it is something trickly about the brain that we perceive infinite DOF when we look at things, but... of course, the eye is ultimately just a kind of lens which does have finite DOF.
There are many interesting perceptual oddities, not just in sight but analogously in sound as well.