RE: CASWELL HAZARD & CO. GINGER ALE NY Round bottom
You've done some good research- your bottle is probably from the 1875-80's. The aqua Caswell Hazard is not one of the rarer American round bottom sodas. They also come in a greenish aqua which is harder to find. The aqua variety comes up at bottle shows fairly often. In the near mint condition yours seems to be, it is a $20-25 bottle on a show table. What might happen on the auction scene, well, you never know. I've seen super rare round bottom bottles go for a song, and then the common ones are through the roof. There are quite a few American round bottom bottles, many that are embossed ginger ale. Many had no embossing, just a paper label. You have to be careful about round bottom bottles with labels only. There are too many labels and plain round bottom bottles that get together that shouldn't be!
As far as not finding anything on Caswell Harzard selling ginger ale, that is unusal, as ginger ale was so popular in those days most bottlers and pharmacies (I think Caswell Harzard did have a storefront in NYC) always promoted their ginger ale. And Caswell Hazard was, as you pointed out, actually quite a big company, even for those days. They sold products everywhere they could. That's why diggers find Caswell Harzard item in many parts of the country. Ginger ale had a lot of reputations, but the primary one was not medicinal. However, because it was made with real ginger it did have medicinal benefits for the digestion.