I have two wooden chairs that are coming apart and I want to glue them back
together. The rungs/cross-pieces that connect the chair legs together keep
coming out, which makes the chairs unstable.

In the past, I have used wood glue (carpenter's glue) to fix other wooden
chairs, but that never seems to work very well.

I was just wondering if Liquid Nails, or Contact Cement, or some other type
of glue might work better. My reason for wondering about Liquid Nails or
Contact Cement is that I think they both remain slightly flexible/elastic
while wood glue becomes brittle and seems to crack easily.

Anyone have any ideas or suggestions on what might work best?

Thanks.