I am an owner. If I have a suspect chair, I remove it. There would be liability if a guest sat in it and it broke. My housekeeper or I check the property and the furnishings after each guest, and I as the owner would not charge a guest for a "broken" chair unless I found it splintered all over the floor. Furnishings do wear out or become suspect after a certain amount of use, but I would not charge a guest for a loosened leg.
If you broke it unintentionally while a guest, and you knew that it broke due to your use (or misuse), I would expect you to contact the owner immediately on breaking the chair and offer some amount of compensation.
I am siding with you based on your description of the circumstances.