There are metabolic processes in which ATP is synthesised without the involvement of ATP synthase. The best examples are, in fact, two steps in the glycolytic pathway, catalysed by phosphoglycerate kinase and pyruvate kinase. This is why, in the absence of any aerobic metabolism, many organisms (like yeast for example) can grow quite happily, producing two molecules of ATP for each molecule of glucose that is metabolised.
Presumably this process is what enabled ATP to gain its central role in energy metabolism without any need for an ATP synthase.