I'm not a golfer but my best friend is a very good golfer and he keeps me informed of the state of the golf "industry". He says a major problem public (and some private) courses face is unrealistic expectations by golfers. Golfers watch tournaments on tv and see very expensive courses that have been coddled into near perfect tournament condition and they set that as their expectation of how all courses should look and play. Courses are astronomically expensive to maintain, even courses that are not in great shape are expensive to maintain. Labor costs alone can easily run into many hundreds of thousands of dollars a year at a small course. Equipment, fertilizer, pesticides, herbicides, carts, water (if a course doesn't have an adequate on site water supply), fuel for equipment, etc. all add up to make operating a golf course at any sort of profit a daunting task.
A private club with 500 members paying $600 a month in dues enjoys $3.6 million in revenue from dues alone every year. They also have revenue from food and beverage, pro shop sales and guest and cart fees. Even with those revenue streams, most private clubs are lucky to break even each year and frequently have to turn to assessments for capital improvements. Public courses rely on green fees, cart rental fees and what little clubhouse sales they can muster to try and show some sort of profit. I think it something of a minor miracle that any public courses are able to stay open.