Bobby Jones as a putter is a tough question. He was obviously very successful, but you are correct that his technique appears odd today.
One point worth noting in assessing Jones' putting is that his main competitors of the day were Walter Hagen and Gene Sarazen. Hagen himself was a great putter, but Sarazen really was not. Another point is that Jones had two main phases of his golf career -- the Lean 7 Years (1916-1922), and the Fat 7 years (1923-1930). He retired in 1930 as an amateur, so his career sort of blazed across the sky with a fantastic intensity. He won something like 13 majors out of 21 he entered -- a pace or percentage about 700% hotter than Nicklaus' pace.
During the Lean years, Jones missed many short putts and had quite a temper. After making a great showing at Merion for the US Amateur at the age of 14, Jones was an instant superstar child prodigy and was expected to win many tournaments immediately. But he really didn't start winning majors until he was 21, when he won the US Open. He received a putting lesson in 1924 from Walter Travis, America's first great amateur champion, and then went on a winning binge that has not been equalled since. Travis adjusted his stance to get his heels close together, and made his stroke more flowing with some wrist action. Jones (and later Ben Crenshaw) described Jones' stroke as a miniature goilf swing. Francis Ouiment, the giant slayer who was the first American golfer to win the US Open in 1913, described Jones as "a demon" on the greens.
Hagen, who was America's first pro golfer superstar, won five PGA Championships and was the undisputed master of match play. In a challenge match against Jones in Florida after Jones had attained his career heights, Hagen trounced Jones.
Since Jones was absolutely astonishing with his driver and iron accuracy, bot for line and distance, it is no surprise that using a putting stroke that was a miniature golf swing also allowed Jones to perform miraculously on the greens. While on a day-to-day basis, it is probably true to say he stuck his approaches so close so routinely that his putting skills were seldom challenged, it is equally true that on those occasions when he was faced with a 20-footer to save par and win a tournament by one stroke, he was a very cool customer and came up with the goods when he needed them. he was known in several majors for sinking MONSTERS of 90 feet, 120 feet, 150 feet! To me, this simply testifies to his extraordinary sense of distance, space, timing, and movement.
There is no question that Jones was a superb putter and that he was one of the very best of his era.
Horton Smith, the winner of the first Masters in 1934, is a fitting successor to Jones as a putter, and probably surpassed him technically.
Jones learned golf at the Atlanta Athletic Club's East Lake Course. This course was built when jones family first visited there, when Bobby was five, in 1907. The architect was Tom Bendelow, a Scot who equally Donald Ross in the number of courses he designed. Ross actually renovated this course in the late 1920s. The course featured two greens for each hole, with the summer greens being bermuda and the winter greens seeded with rye. In 1963 George Cobb redid the course and converted the greens to bent.
It is not quite correct to think of America as having to convert from sand greens to grass greens. The first courses in America were totally grass, such as the Orchard in NY. In fact, the Rules of Golf did not formally recognize a "green" as a distinct feature of a hole until 1956 or so. before then, what is now the green was just an extension of the fairway, and had the same grass.
I'm not quite sure where sand greens originated or when, but it was probably in some arid region of the British Empire in the late 1800s. Sand greens are used wherever water is scare or unavailable for irrigation and the climate will not support grass cover all year, especially during the summer. Pinehurst had sand greens during the 1900-1925 era not by choice so much as by reason of climate and difficulty of irrigation. Sand grrens were very flat, and the Pinehurst greens were also square or rectangular in perimeter. The sand was coated with an oil, and had to be specially raked after each putt. All putts were completely straight! And the hole was seldom if ever moved.
In the 1920s, there were a number of people developing grasses suitable for greens in dry climates. Also, creeping bentgrass as a greens turf, was first developed in 1927. Donald Ross converted Pinehurst greens in about 1928 from sand to bermuda. Augusta National, built in 1932, featured bermuda greens. As better heat-resistant bentgrass strains were developed in the 1970s and 1980s, many bermuda greens in the south were converted to bent. Pinehurst went bent in the 1980s, and Augusta went bent in 1981.
There were many sand greens in early America, 1900-1940, especially where heat and aridity is a problem. Miami and across Florida had sand greens for a long time. There are still some sand greens in the US, and one source says that North Dakota alone still has 50 courses with sand greens.
Sand greens also remain in Australia and Canada, although obviously they are no longer the norm even for harsh climates. Even Dubai in the Arabian Desert and Morocco in the Shara Desert have lush greens today.
I've created a web page for Bobby Jones and another for Sand Greens that you might find interesting.
Sand Greens: http://puttingzone.com/sandgreens.html
World's most advanced and comprehensive putting instruction.