It's not true to say that all modern glues are flexible. The feeble white PVA wood glue that you can buy in DIY shops remains flexible- that's true.
However, Marine chandlers will sell you two pack epoxy glues which penetrate wood deeply and set iron-hard.
There are some excellent Acrylic single part glues too. I use one which sets to a glass-like brittleness, and another which includes a foaming agent which is fantastic for filling gaps but remains slightly flexible.
The rabbits are safe from me!
Cheers
Dave