Tom
I think you have two basic eras, here:
--The early age (1880-1900), when Morris and Park monopolised the laying out of new courses/altering of very old ones.
--the later age (1900-1930), when Braid (and to a lesser degree, Colt) built other new ones on lesser ground and modernised many of the courses that Morris and Park laid out.
I don't think that Colt is in the running for either question, in that while his work was exemplary (particularly Eden and Muirfield), it was limited.
I think that Morris' work had the most influence because it was the most extensive, and at the better venues. Regardless of whether or not one thinks that much of this work is NLE (or at least hidden to the undiscriminating or 2nd-hand eye), you can't argue with the pedigree of the tracks that even today are proud to credit him with their design.
As for the best work, Colt might win if he had more chances to show what he could do at situations which weren't the slam-dunks that Muirfield and St. Andrews were. So, I'll vote for Braid, whose stuff is consistently good and occasionally brilliant. Very much the Donald Ross of Scotland.
