I'm not an expert on steel and don't claim to be but I am a machinist who works with many exotic metals and loads of stainless. I have considered making my own stainless axles, this would take very little time, but safety wise the choice of material is critical, I prefer to err on the side of overkill. Most stainless steels that I would consider using for an axle are very expensive, heat treated 400 series or 17-4PH. I have access to 316 coming out my ears, but the tensile strength worries me, it's less than low carbon steel. 316 is essentially the same as 304 just better corrosion/chemical resistance.
17-4PH would be my first choice as it has a strength comparable to 4140, but my cost on that is not worth it to me.
Stainless does work harden but in this usage would not be an issue I don't think as there's anything really running against it.
4140 or 4340 would be fine choices as they both have excellent mechanical properties, but you have to deal with rust unless you have them treated. These are not the type of steels that you would use for through hardening typically so don't worry about heat treatment, the bars are usually HTSR from the supplier.
I suspect that the originals are some sort of plain carbon steel that has been case hardened, it's highly unlikely that a factory part is made of high carbon steel. In an axle application I would suspect that shear strength would be the ultimate design property. I may take one to work now and see exactly what it's made of as we have the machine to check this, I will not try to name it as I don't know.
And finally any machinist worth his salt does not leave stress risers in a shaft of any kind, grinding should not be neccessary, a slower feed rate with a radiused tool will do fine.