Griffith Park - Wilson seems to have fallen off your list (if it was ever on) and I think you should consider including it. It was designed (or redesigned) by George Thomas in 1923, hosted three Los Angeles Opens (1937-1939), and was considered by some to be among Southern California's best courses. (See blurb above, for example.)
On the eve of the Los Angeles Municipal Championship in 1927, the Los Angeles Times had this to say:
The Woodrow Wilson Course, one of the very finest municipal layouts in the United States, has been carefully groomed for the championship opening Monday.
(Those signed up to play included top golfers from all the area clubs, with a particularly strong team from Lakeside featuring George Drugstore Cornes, Howard Hughes, and Jack Neville.)
A 1938 Times article announced that the Griffith Park courses would again host the Los Angeles Open in 1939:
Held the last two years at Griffith Park, the tournament drew the largest total attendance in its history with Southland fans flocking by the thousands to the beautiful public courses, said to be the finest municipally owned links in America.
(Competitors planning to play included Eddy Loos, Joe Degal, Jimmy Thomson, Sam Snead, Horton Smith, Ed Dudley, Lawson Little, Byron Nelson, Harry Cooper, etc. Prior hosting courses had included LACC North, Riviera, and Wilshire CC, and the tournament returned to LACC in 1940.)
Take it for what its worth because he is certainly no Mike Cirba, but here is what Daniel Wexler wrote about the Griffith Park Courses in 2007:
Griffith Park, Harding and Wilson Municipal Golf Courses (Los Angeles): Though golf made its debut in Griffith Park in 1914, today's facility dates to a 1923 expansion to 36 holes by the legendary designer George Thomas — a project that the philanthropic Thomas reportedly augmented from his own pocket when city funding ran short. The site of three late-1930s L.A. Opens, as well as the longtime home of 1961 PGA champion Jerry Barber, Griffith Park once ranked among the world's finest period municipal golf facilities. Today, after decades of ill-advised modifications (e.g., Harding course's now-removed "water bunkers") and the building of the Golden State Freeway, it remains among the most historic.
Don't get me wrong. I am not claiming it was the absolute best public in the country -- such a claim would be asinine-- I wasn't there so how would I know?? But surely it deserves consideration to be added to the list . . .