Aerodynamic theory has its origins in hydrodynamics developed to make better ships centuries ago. Most of hydrodynamics applies to aerodynamics and served as a good starting point. Air is compressible above M=.3 but it did not matter in the early days because nothing went more than about 25 mph.

Theory and practice went along parallel lines historically. The Wright Brothers pushed it along experimentally, Otto Lilienthal before them and many others contributed to the body of knowledge to make it the bonanza it is today.

As far as whether early experimenters were able arrive at estimates of lift without fully developed theory, I am sure they were but it was time consuming, dangerous, slow, and expensive. Otto Lilienthal died in one of his own experiments. Ideally theory and experiment go hand in hand. Often someone will intuit a principle while theory only follows later and sometimes not even from them. It can also go the other way and theory can drive a discovery as well. For example, Buseman and RT Jones both theorized the benefit of wing sweep in the early 1930's but the benefit wasn't realized until the age of jets after WWII.

As far as infinite wings are concerned, all you have to do is put plates at the ends of wings in a wind tunnel to simulate an infinite wing. The wind itself tunnel can be the ends of the wing also. You will not find vortex rolls like those seen in real life aircraft. However, all the theory involving vortices still applies. The idea of vortices in lift is far more than just those wingtip rolls you refer to. It forms the basis for all modern theory of lift in the low-speed realm.

Skipper- yes I studied aeronautical engineering and have the t-shirt to prove it .