I would use an abrasive and get all the old glue out. A small flapper sander should do the trick. Or wrap an undersized dowel with self-adhesive sand paper and use that to clean it out.
The problem with that is the fact that in doing so, the hole will get bored out larger and the spindle will fit sloppily into it, because the wood glue penetrates into the wood. So the only way to get it all out is to remove some of the wood. There are ways to deal with that situation, but that's not the OP's question.
It depends upon what you consider to be of greater importance.
1. Old glue residue that might compromise adhesion.
2. A subtly compromised fit with good adhesion.
I think the second choice is a better option. The there are going to be multiple rungs to lock in the uprights. The rungs are not being stressed in tension. They are going always to be stressed in the downward direction. So if the rungs are clamped in that direction the 70% or so contact will make a strong joint.
We don't know the original adhesive. It might have been assembled using hot glue, in which case the wood glue might not hold at all. He could wedge the rungs in the holes, but If I were worried I would cross pin them with 3/16" or 1/4" dowels (or even nails).
A dowel wrapped in sandpaper can be directed at the offending glue. If it is no longer a tight fit, then the best gap filling structural glue is probably epoxy. Gorilla glue gap fills well, but that fill is not structural.
So I agree with the epoxy if the fit is compromised to the point that it is no longer a snug fit. But I would clean out the old glue no matter what. Even epoxy might not adhere to something like hot glue, which is more wax like than anything else.