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?

Yes. Response speed for both autofocus and shutter release.

The mirrorbox has a number of side effects that aren't self-evident. Like the ability to use a completely separate autofocus sensor array. dSLRs, for the most part, do NOT use the main image sensor for autofocusing, the way mirrorless and compact digital cameras do. Phase detection autofocus sensors are in a completely separate array on the floor of the body and the mirrorbox is actually used to direct some light from the lens down to that array as well as up into the viewfinder.

Mirrorless and compact digital cameras tend to have additional shutter delay because composition must be done through liveview, and to avoid a ghosted image, all charge must be cleared from the sensor before the main exposure is made. A dSLR's optical viewfinder doesn't require this. With the mirrorbox and mechanical shutter in front of the sensor, the sensor itself does not need to clear any residual charge before taking an image unless liveview is used. This increases shutter responsiveness.

While there are strides being made in introducing phase detection from the main image sensor and shutter delays are reduced, dSLRs are still the tool for choice for fast action photography. Tracking autofocus performance, and autofocus speed are still better with dSLRs.

In addition, the use of older film-era technology also means compatibility with film-era gear. dSLR cameras can typically use film-era lenses in the same mount system with full native compatibility (including autofocus). Mirrorless cameras, while they can use adapted manual-only lenses with limited function, typically only have full autofocus function with lenses in systems that are only 5-8 years old. Nikon and Canon dSLRs are still part of the largest camera systems in existence, with the largest number of native-mount lens choices.