...The way Nature intended
Really, why is there any OTHER method of gardening? Why use chemical fertilizers? Plants grow and gather their nutrients from the soil. Dead plant matter returns to the soil, creating those nutrients. Why spray chemical pesticides? There are many insect-resistant varieties, and while you cut losses by a percentage, and increase yeild, those poisons are absorbed by the plants and then consumed by us. Constantly growing without returning organic matter to the soil depletes and makes it useless. Pesticides do not target specific insects, but poison everything...including beneficial insects like bees, ladybugs, and earthworms, and both fertilizers and pesticides that run off into our waterways are toxic for aquatic life.
We choose to garden organically because it is healthier for the Earth and environment, and healthier for us. Organic gardening takes a little more work and effort than chemical gardening, but it is well worth it, and a wholesome experience. We have an abundance of composting materials and manure on our property, but they are not hard to collect for those who are not as fortunate. Pick up those paper sacks of leaves around the neighborhoods that people throw away! Find a local small farm or livestock yard for manure! Ask your friends and neighbors to save vegetable scraps for your compost! Even container gardening on an apartment balcony can be done organically.
And...picking bugs off plants by hand isn't so bad. If you are THAT squeamish about insects, then maybe gardening is not for you. Nature and plants could not exist without bugs. Get over it!