It seems that sautéing onions is the first step in just about every savory dish from a wide variety of cultures. How did this process come to be so widespread?
Because it is the most common aromatic vegetable in the world.
In most cuisines dishes are made by first cooking aromatics to release their flavors; this is referred to in culinary terms as a flavor base. Then the other ingredients are added in, building a dish. The flavor base you use is what gives regional food its distinctive qualities: Chicken cooked with a French flavor base will taste quite different than chicken cooked with a Mexican flavor base, or an Indian flavor base, even though they are all chicken dishes.
Aromatic vegetables, which give off deep, well-rounded flavors and pleasant aromas when cooked, are the core of flavor bases. The classic French flavor base known as mirepoix is a combination of chopped onions, celery, and carrots. The Italian soffritto varies from region to region, and may be as simple as chopped onion and garlic, or it may include things like tomatoes or pancetta. A typical Indian base mixture for a curry may contain onion and garlic, hot chiles, and chopped ginger.
Sautéing is not always used. In European cuisines the aromatics may be sweated instead of sautéed. In Mexican cooking many dishes have the cook dry-roast onion, garlic, and chiles as the first step. Thai cooks make one of several types of curry pastes for their flavor bases by grinding together aromatics like shallots, lemongrass, chiles, and kaffir lime leaves.
The reason onions are so widely used in flavor bases is because they are one of the oldest cultivated vegetables we know of. They grow well in many different climates and soils, and are easy to store, especially over the winter. Their pungency provides flavor to dishes even for people too poor to afford more costly herbs or spices. These qualities have led them to becoming an important part of the human diet all over the world.
Wild onions grow on nearly every continent, so it is unlikely they were cultivated in one place only, and then spread by contact between various civilizations. They were more likely first used wild and then later cultivated in multiple times and places, which would have contributed to them becoming so universal.
You may be interested in also trying /r/askculinary. Though, to be honest, I do not know how good of a resource they are.