Brass is chosen here for electrical reasons, not because it's strong.
Substituting stainless on an earth rod isn't a good idea because two different metals in contact form a battery and corrode, fastest when the joint is damp. While a stainless fastener will remain physically strong, corrosion between it and the earth rod will become a very poor conductor – the 'earth' is likely to go high-resistance, useless for safety purposes.
How quickly a stainless bolt reacts with Brass depends on what sort of Stainless it's made of, which is unknown. Some are better than others. But it's worth remembering that electricity is powerful enough to break water apart. When two metals are in contact best to ensure they are close together in the electrochemical series. Unknown engineering is bad engineering.
While brass isn't as physically strong as steel, it's certainly fit for purpose in this application, ie connecting a copper earth wire to a brass rod, or a copper-plated steel rod. The most likely cause of failure is stress cracking due to over-enthusiastic tightening. Another possibility is acid damp dezincifying the brass, but the characteristic pink colour is hard to miss.
As Brass is about 80% as strong as mild-steel, the tightening torque should be reduced accordingly. A stainless or high-strength bolt could be 3 times stronger than mild-steel; in comparison a brass nut and bolt is easily damaged.
Failed to find a table of recommended tightening torque for Brass Fasteners, but did find a suggestion that a sample brass bolt should be deliberately broken with a torque wrench, and real ones tightened to no more than 60% of that value.
Dave