While that big blue blob isn't all that informative, believe it or not those lines were already at an alpha of 0.3. Should have been reasonably transparent with any reasonably number of overlaps. But there's just an awful lot of possible 30 year intervals if you look at every possible start month from January of 1871 onward and those settings clearly weren't cutting it.
I played around a bit with both line width (wider) and transparency (much more transparent) and I think this version (while looking a bit blurry) does a better job of representing where portfolios tend to more commonly go.
Some sort of kernel density style transformation -- or a heatmap? -- might work better, but I don't know how to do that in a way that would look correct for time series data.