And with a digital camera, you don't need a hinged mirror as you can show the user exactly what light will be captured by just routing the sensor output to an LCD display.
This is the reason for the rise in popularity of mirrorless interchangeable lens cameras (MILC). Without the mirror box, the camera can be smaller, lighter, less expensive, etc.
Having a mechanical component that needs to be able to move around in a very precise manner and can break or fail seems like a very large liability.
Probably not as much as you'd think. These same companies have been building SLR's with mirror boxes for decades, and they've gotten pretty good at it. There may be occasional mechanical failures, but at this point the mechanisms tend to last a lot longer than the useful lifetime of the camera. In other words, customers will want to replace the camera for other reasons (e.g. better sensors, more features, etc.) before the mirror mechanism fails.
Why do manufacturers continue to build SLR mechanisms into their digital cameras, particularly at the top-end of their product lines?
The main reason has to be that it's what customers want. DSLR's evolved from film SLR's, and photographers still want to buy cameras that let them see what they're shooting through the lens.
Why do photographers seem to heavily prefer DSLR cameras over digital point-and-shoot models that offer the same features but without the SLR mechanism
You answered this yourself pretty well: It let you see through the viewfinder exactly the light that would be passed on to the film. Assuming you care about accurate photography, that's kind of a big deal.
If you're not looking through the lens, you're seeing some digital interpretation of what the scene looks like. Electronic viewfinders (EVF) have improved tremendously in recent years, and they do have the potential to show you what the sensor will record, but that's not the same thing as seeing what's visible through the lens.
(there are, for instance, full-frame point-and-shoot interchangeable lens cameras available, though it's not clear if they're very popular among photographers), to the point where "DSLR" is almost synonymous with "serious photographer's camera"?
There are definitely "serious photographers" who have switched to MILC's. David Hobby and Zack Arias are two examples of well-known photographers that use Fuji mirrorless cameras. However...
There's a lot of inertia that will need to be overcome in order for DSLR's to really lose popularity among professional photographers. Lenses present a huge obstacle -- photographers already have large investments in lenses, and manufacturers have large existing lines of excellent (and profitable!) lenses for their DSLR lines. If photographers start jumping ship from Nikon or Canon to Fuji or Sony, and if Fuji and Sony can provide lenses that the pros need, then Nikon and Canon will surely start producing more top-end lenses designed for (and not merely adapted to) their mirrorless lines.
Is there any significant benefit to having an SLR mechanism in a digital camera? Particularly in terms of a benefit that's large enough to make up for the liability of adding a mechanical part into a design where a solid-state alternative is available?
Again, I think you may be overstating the liability of the mechanical system. These things work really well. So, let's turn your question around and look at the other side: Is there any significant benefit to changing the reliable and well-understood DSLR design? Obviously, the answer is yes, because MILC's are getting some real traction in the marketplace, but at the same time the answer is not YES!!!, probably because the liability is not as great as you imagine.
My own feeling is that a much more interesting question is: Will DSLR's ever get electronic shutters that could give them much higher flash sync speeds, faster burst modes, and faster shutter speeds? I think you'll see that happen before MILC's unseat DSLR's.