I’ve experimented with hydrogen.
The welder you describe uses electrolysis, to split the hydrogen and oxygen from each other. Problem is...it takes electricity to do this....and it is only 40% efficient.
The experiments I tried were to basically use spare battery power to create a small amount of hydrogen, and vent to the intake (hydrogen ‘boosting’)
I actually got worse gas mileage, since the O2 sensor sensed too much O2, and compensated by pulsing more fuel in.
This can by fixed by reprogramming the car’s computer...which I can’t do. There are commercial kits which do this. However, I can see problems with the water freezing in the winter; and, the plates used in the electrolysis (stainless steel) immediately have a coating form on them...which would have to be periodically scraped off.
All in all, not a great idea (hydrogen ‘boosting’).
And actually using water and battery power alone to power the car? Won’t get very far - alot like using a fan to power a windmill. So don’t worry, no conspiracy to be found.
One option experimented with in Japan is home fueling stations for hydrogen. Essentially something very similar to that welding box is converted to fuel up a hydrogen fuel cell...and this can be done in your garage. The problem comes back to the efficiency problems of using electricity to get hydrogen...and this is less economical than gasoline.
Thanks for the info!