I don't venture away from Rustic too much, but that may just be me. LAX is about 50 miles from Rustic. Many of the non-true public options you will hear about here are that far or further.
The trouble with the true publics is that they are extremely overcrowded, slow, and have generally been watered down both figuratively and literally. Of these Rancho Park is probably the best city course close to LAX, and as mentioned Long Beach and Los Verdes are close and might be worth checking out on a weekday and/or at an odd time. Santa Anita is a drive but it is a good county course, despite overcrowding and less than ideal conditions. It is a "template course" in a place one would never expect, a very interesting WPA-type course on formerly flat land which was rtansformed into something else, and a neat story. (I like to say in partial jest that it is like the old Merion legend, only true.) The Brookside courses in Pasadena are also not horrible but not great either. Generally too flat and dumbed down over the years, and not convenient to LAX.
Of the newer daily fee there are many more options but they are generally more expensive and at least as far or nearly as far as Rustic.
- To the west toward Rustic are the Lost Canyons courses , which are two severe Dye-style courses in some land that looks cool but isn't necessarily suited to golf. Some good holes and good sections and some dramatic holes and then a lot of really bad holes. Cart golf. A terror in the wind, but might be fun if you enjoy pain. Also out that way is Moorpark CC and Tierra Rejada, both abysmal in my opinion.
- To the north is Angeles National by someone who works for Nicklaus. It might be walkable except that carts are (or were) mandatory. To my mind the course looks better than it plays. Lots of environmental areas, stuff that doesn't really make much sense, but again some people really enjoy it. Further out isRobinson Ranch, which is two Ted Robinson courses known for fast greens and fast greens. Sections of each course are decent but there may only be a handful of good holes among the two courses. Again though it can be enjoyable golf and some like it. Also there is TPC Valencia which is worth seeing only to see how bad a course can be.
-To the east are some decent Curley- Schmidt courses, the cheapest and maybe most interesting design is Goose Creek. Many fun, strategic holes but the site is very flat and the place smells like goose crap. Oak Quarry has some dramatic holes but is sort of a mixed bag. Oak Valley hard, some very good holes, Curley Schmidt in Pete Dye style.
There are more than these but that is what comes to mind right now. Don't want to get you too pumped up with these exciting prospects.