I have to agree with Leon. Elmers Glue Stick should work just fine. When the labels were originally put on the bottle no one cared what glue was used they just put it on the label and slapped it on the bottle. There are bottles out there that have had there labels for over a hundred years with no problems with the glue destroying the label. I doubt if modern glue, such as a glue stick, made for applying paper to other objects is going to do any damage.
You're right. People didn't overthink things back then. My name is Curt, not Leon though.
Acid-free rubber cement, which Elmer does put out, can be had in most any hobby section of stores. Use it and it only. Other glues may bleed through, alter the colors as they react with the inks, leach out acids, or otherwise crumble in time.
Rubber cement can be safely removed if desired but maintains a strong, flexible hold that in no way damages the paper as it is inert and stable throughout the years. Archivists use it, from book restorations to backing antique paintings or photographs.
I use nothing else but acid-free rubber cement for reattaching labels. Clear going on and when dry, stable, flexible so it bends instead of cracks when impacted, removable, etc.: The perfect solution.