<<In a good weld, the weld joint goes fully through the spliced area. In that case, you could weld two pieces of flat steel together at the ends, then grind away the surface weld on both sides and have the appearance of one flat piece of steel, without diminishing the strength of the joint.>>
The overly simplified explanation is that welding is more or less a cast-in-place operation. As such, grain structure within the weld is very different from that found in the worked material being joined (likely hot or cold rolled in your example). In general, cast properties are a little lower, which is why you need greater cross section in the cast area in order to equal the rolled material. Given equal area (weld ground to profile), the weld is the weak link.
A great many practical structures are designed around stiffness, not strength, and don't stress their components anywhere near maximum. Easy to see how someone with lots of practical experience in general fabrication might believe grinding welds has no downside. With aircraft structure we push closer to the limits of material strength because we're always concerned with weight...so we gotta be a bit more concerned with details.
