I think CSAs are a skill, actually. They can be frugal, but obviously not if you don't know how to use them or don't like seasonal produce.
Yes, CSAs will give you just greens in the first part of the season. That's just biology, actually. Many people may criticize that, but all plants start the season by growing leaves. Only later do they develop their fruits or tubers. We do get a few special treats like asparagus and radish and green garlic. It's the season of greens.
This is my third year with a CSA. If it's going to work for you, you have to adapt to it. You can't expect it to fit unobtrusively into your diet if you're just looking to replace your frozen corn kernels or your can of green beans. It won't leap out of your fridge and start stir-frying itself.
I think the most success with CSAs comes when you see it as the central part of your diet. Instead of going "will we have chicken or beef tonight?" you're thinking "ok, I've got kale, bok choi, radishes, green garlic, lettuce, (etc)... I could do a stir fry, or maybe an omelette or souffle or a quiche would be good, or maybe some curry would be nice." Then you add whatever else you have to that: "I could put chicken in the curry, or maybe just chick peas." Or "The cube steak I have will be perfect in the stir fry." You start with your veggies. If you just do your meat/veggie/starch dinner and are just looking to replace your frozen corn kernels on the side, it's just not going to fit.
In three years, I have had very little produce go bad on me. Maybe three items?
Eating seasonally has been a simple requirement up until the recent petroleum age. Our ancestors would not be able to relate to us not even knowing what season things are available in, or how to deal with food that is only grown in a certain season.
It's hard for me to imagine how the food might not fit together. The food harvested in a given season has fit together for a long time. Only recently have we had recipes that call for asparagus and butternut squash together (seasonal gymnastics). Seasonal food fits together very nicely. Apples and acorn squash. Tomatoes and garlic and basil. Eggs and asparagus. Carrots and potatoes and parsnips in a thick stew.
CSAs aren't magically frugal by themselves, you need to develop the skill to make them so. The produce is just a tool; you have to actually wield them. You could say you might save money by doing your own car tune-ups, you can buy the tools, but it won't happen unless you learn how to tune up your car and then actually do it.