The only time a connection to earth serves a useful purpose is when there is a short in the power supply. About the only thing the connection to ground will do is providing an alternate path for the current to "ground" so the fuse or circuit breaker will trip. In theory it could save the welder from becoming a light bulb for a second or two before dropping to the ground dead as a mackerel.
There is a flip side to every story. Failure to provide for a proper earth connection can potentially provide a job opportunity for the next welder looking for a job. It is just another case of Darwin in action. Only the smart survive to earn a respectable living.
It reminds me of the time many years ago when I was visiting relatives in Florida. My cousin wanted a bumper hitch installed on his car. He asked if I would be willing to do it for him. I said it wasn't a problem if he would lend a hand and if he knew someone with a welder.
He said, "No problem. My buddy's got a welder in his garage."
Off we went. I had Roy holding the part against the frame as I tacked it. As soon as I struck the arc, Roy let out a yelp like a 12 year old girl!
"Come on Roy, it can't be that bad! It's only 60 or 80 volts if that." I told him.
He said, "It really bit me!"
"Roy, the maximum OVC is about 80 volts. These little buzz boxes usually put out between 40 and 60 volts. So stop whimpering and hold that piece up." was my retort.
Once again I struck the arc and again Roy let out a howl like a little girl getting a flu shot.
"Come on Roy! What's the matter? I asked him.
"I'm getting my multimeter. I'm getting a heck of a lot more than 40 volts!" he muttered.
Roy checked the voltage at the terminals. He was right. He was getting wacked with 230 volts. Luckily he was lying on dry concrete. As it turned out, whoever hooked up the welding machine hooked it up wrong.
Best regards - Al