We can make informed guesses. For instance:
In steady travel, the hovercraft are clearly less far off the ground than their width, based on the width of the shadow versus the height between shadow and craft.
Let's assume, to be conservative, that they're twice as long as they are wide, and that their distance from the ground is the same as their width, and that the height of the craft is negligible.
We can see the tail of the shadow on the ground, so if the Sun was directly ahead and inclined a bit above the horizon...
...It would have to have a run-over-rise of at least 2 to knock the shadow off the screen to the bottom. That implies an inclination of about 28 degrees or lower.
Now, this is the worst-case scenario for the Sun pushing the shadow off-screen. Much of the time the camera is pulled back, and at other angles the shadow would usually be visible at much lower inclinations (for instance, if the Sun was instead behind the camera, it would basically have to go down for the shadow to disappear).
If you trace a cone straight up into the sky so that its edges are at a 28-degree inclination, that cone
will subtend 3.33 steradians. A hemisphere is 2pi steradians, so even in this cartoonishly conservative estimate, a correctly-projected shadow would be visible more than half the time during the day.
Under direct sunlight it borderline doesn't as far as the naked eye is concerned. I'd snap a picture under true sunlight, but since it's night, here's an example from the Ultra Sophisticated Closet World Simulator With White Walls To Produce Lots Of Ambient Light From a Strong Directional Source Such as a Lamp Sitting On The Floor:
AO is clearly visible in places like the shadowed box crease on the right side, but what little dimming you get from the small fraction of the hemisphere blocked by the battery is utterly tiny compared to the intensity of the direct light source.
It's quite clear in the video, and must be in order to aid the player's sense of location.
The reason it doesn't look that out of place is that FRN isn't dead set on strict realism.
(And also probably that our senses are somewhat dulled, as the number of games that have mostly correct-looking shadows over large environments without aliasing or bizarre artifacts is basically zero.)
Hmm, link to what exactly they said? There are reasons to use SSAO instead of a baked map
even if dynamic objects don't interact with it, like storage/memory considerations.
Shin'en doesn't seem like a group that would do something "just because" when they could bet equally effective results in an entirely cheaper way.