Simple really; the Earth does _not_ in fact do the things that you suggest. _Grounding_ an energized case will leave that case still energized, an earth electrode will make a _very_ poor neutral, and a welder will work just fine without any connection to earth at all.
1) The 'grounded' case is made safe not by the connection to earth, but by the connection back to the electrical transformer neutral. In the event of a hot to equipment ground fault, the circuit breaker will trip, removing the hazard.
2) The midpoint of a 120/240V electrical supply is from the metallic connection to the transformer neutral, _not_ the connection to earth.
3) The 'ground' connection on a welder is simply one of the welding transformer terminals, the one that by convention is connected to the bulk material, not to the welding terminal. Since the bulk material is often in contact with earth, this becomes the 'grounded' transformer output.
Search this forum for the terms 'grounding' and 'bonding', and you will find many discussions of the topic.