We welded the shaft of a flail mower (brush hog to you folk I think), and it's held up for 15 years, A good weld is darned strong.. I would recommend the Arctec 220 XC rod for an arc welder, 85,000 lb tensile strength and flows nicely... One of my favourite rods, 7018's are very similar (including the lousy restrike characteristics) and would probably do pretty well too. When I did the front end conversion on my truck, I had to cut and weld the drag link on the steering and used that rod... perhaps not quite legal, but I figure the accident will have already happened if that thing breaks!
When we welded our mower shaft, we didn't have a lathe, and we just set the shaft in some wooden V blocks and spun it by hand, and used an angle grinder to clean it up, final with a sanding disc. Then we had to cut a groove into it for the key and did that with the grinder too.... it's worked for years now