What is the best (cheapest/quickest?) NEC compliant way to join two
wires within a wall that will NOT later be accessible without
disturbing the building finish?

There is no way to do it without being accessible. Look at the fiasco of
things like aluminum wire for a good example. Any time there's a connection
made you need to have access to it. As has been suggested, pull the wires
into a single gang junction box and put a wall outlet in it. Accessible for
any future needs and potentially useful in the room. Otherwise just put a
blank faceplace on it.

The current method I most often see utilized in past construction is to
strip an area of insulation off of a wire without cutting it, twist
another wire around this wire with many loops and finally apply
electrical tape generously.

Then you're not looking at something that would pass code.