I have a wooden chair with four legs that fit into the seat, with front and
back legs on each side held from splaying out by a fore-and-aft cross-brace;
those two braces are themselves joined by a side-to-side brace. On one side,
the braces have come loose from the holes in the legs, making the chair
rickety. I tried gluing them with PVA glue but the glue joint broke almost
immediately (after letting it set for several hours).

I don't want to have to unglue and re-glue any of the other joints (in case
I make things worse). What is the best glue to use for sticking a joint
where one wooden piece fits into a hole in the side of another? I can
extract the cross-brace by about 10 mm, but there isn't enough play to
remove the brace totally to line the hole with glue, so I have to rely on
straining the joints to expose as much of the ends of the brace, smearing
glue around the exposed part of the brace (at both ends) and then pushing it
back together.