need metal-to-wood glue advice (or other ideas)

[Fumesucker]

11. Put the bolt in the insert and an extra nut on the bolt and run it down against the insert

After you tighten the insert and the nut together you can then just use the bolt to screw the insert in, once you get it all the way in hold the bolt with one wrench and loosen the nut with another then just screw both of them out leaving the insert in place.

Another way to fix that would be drill the hole out even bigger and glue in a piece of dowel that size and then after it dries drill the dowel to the right size and screw in the insert, coat it with some glue when you put it in and hopefully it won't pull right back out. That's going to be a couple of fairly big holes though, if you don't already have the drills it could cost more than it's worth.

Get some Locktite thread locking compound and put it on the screws where they go into the inserts when you install the arms, let it sit for a while and they shouldn't unscrew again.

ETA: I'd tape the chair around the area of the repair too, that gorilla glue makes a bit of a mess if it expands too much, lot easier just to peel off the tape than scrape off the glue.