After some practical field study, I think it is safe to say that the quickest and easiest way to get to Goddard is via Lamarck Col. The route is nearly all on trail and shorter than any of the other routes listed. Just thought I'd mention this because I thought it should be more prominently displayed in your fine list of routes rather than as an afterthought at the end. Another cross-country route that may be better than Echo Col and is the shortest route going strictly by miles, is over Haeckel Col. That route was not field tested and is almost assuredly slower than Lamarck Col.
Pen and I found Lamarck Col preferable to the cc 24 miles of trail from South Lake over Bishop Pass and to Wanda Lake. Lamarck Col might have 3,600' of gain starting from North Lake but gets you to Wanda Lake in only 14 miles.
The Piute Pass approach is very long, I did it in the late 90's and would not recommend it.