"is there any reason to consider this method as a replacement for the compost pile?"

Non that are practical.

I use this to get rid of kitchen scraps. Just cut off the bottom of a 5 gallon plastic bucket and buried in the ground next to an orange tree. Dump vegetable, meat, ... any kitchen scraps into it. The lid (at the soil line) screws on and keeps out racoons. In hot weather it "digests/dissolves" quite fast. In cold weather I use a 20 gallon trash can the same way.

Anaerobic digesters are mostly used to capture the methane gas that can be produced during the digestion process. That gas can then be used for cooking, heating, or sometimes to dry animal manure for sale. In addition to the digester you would need a method of compressing the gas produced and storage tanks, and enough material available on a steady basis to generate enough gas to either replace what you use or maybe even to sell. Few home compost piles will do that.

Thats pretty much what a septic tank does - anaerobic digestion.

Bokashi composting appears to be anaerobic.

There are small(ish) dairy and poultry farms doing a version of this, the methane produced can heat the barns in winter and provides electricity for the farm's operations. I do not think a typical home produces enough waste to keep a system of that kind running.

Here is a link that might be useful: dairy farm digesters

Thanks folks - never thought of my septic tank as an 'anaerobic digester', but it sure seems that's what it is. Does my septic tank generate methane?
Also, why is this proicess deemed 'more efficient' than aerobic composting, at least on a large scale?

As gonebananas mentioned, Bokashi is anaerobic. Its working extremely well for me (in the middle of a large city without a place for a large compost pile away from the house.) Its also very fast. 3-4 weeks from collection to gorgeous compost and *everything* goes in it- cooked food, bones, meat, dairy, fatty things, etc.