You can do the experiment at home but there’s many videos online: place the cork plug on a plate and put in inside your microwave for a couple of seconds. The glue melt.
It would surprise me to find out that the traditional, straight-from-the-tree kind need adhesive to stay in the bottles. I mean corks have been used to stop bottles for hundreds (thousands?) of years; surely they didn't use glue all that long ago?
I suppose - to supply a speculative counter-argument myself - it may be more profitable to cut the natural corks X% smaller, and counteract the mechanical deficit with glue. (That's, uh, kinda typical of the world we live in.)
Can anyone out there speak directly to that?
Regarding the pre modern chemical techniques: not an expert but it seems people used to drink younger wine back then, and used wooden barrels or clay amphora for the older times. For in between times (and the amphora) solid cork as you mentioned seem very plausible.
Considering the material deficit: consider that when you do the microwave cam cooking, the cap double in size and don’t fall down in many small parts. It’s so big it’s basically impossible to put it back inside the bootle. My 2 cents hypothesis is they use glue to use more cork and have a better sealing, enhancing the conservation.