I am sorry to say that I think I don't like that construction. The crosspieces should have a mortise and tenon joint into the leg, in addition to being glued as you say. Maybe they do; I can't tell from the pictures.
The bolts should go into a metal threaded nut inserted into the leg. Ideally there should be one in the crosspiece as well. Without that they will eventually work loose from the wood and back out from the hole in the leg without rotation as your lower picture shows, even if the bolts are held in place to the crosspiece as suggested in other answers. If you can install inserts into the legs your chairs will give you fewer problems. That will be hard with the crosspiece in place and it should have been done when the chairs were made.
I have a table made this way with a single large bolt through the diagonal crosspiece into the leg. It is held there by a wingnut. It's easy to remove the legs for transportation (the top comes apart, too, in a puzzle-like way) and re-assemble the table. I've had that table for several decades and moved it half-a-dozen times.