Let me be the first to state that your b/f is wrong. If he is wrong about putting screws in a chair to repair it, what else could he be wrong about? I'm just sayin' you might want to ditch the b/f and keep the chair.
Never trust a b/f who wants to repair a chair with screws.
Nothing that you have show us indicates that the chair should be gotten rid of. You are correct though, it should be repaired if you really want to keep it.
The splitting of the arms is a fairly simple fix -- a little yellow glue and some clamps and you will be good to go. The trick is to get the glue where you need it. A glue syringe would due nicely.
It may be that the rung that is immediately below the split is loose and needs to be glued as well. If that is the case, you will not be able to use yellow glue (yellow glue does not stick to previously glued parts) and would need to use epoxy, and of course, clams to tighten it and hold it.
If the rung in the picture needs to be glued, you will have a bit of a hectic repair -- most likely both the split and the rung would have to be repaired at the same time, using two different glues.
Yellow glue will not really stick to previously finished wood, so making a mess is bit less of a concern. Epoxy on the other hand will pretty much stick to everything and make a big mess. Have care if you use epoxy. you can protect the finish by using blue painters tape to mask off areas around your fixes. Just pull it off when you are done and nothing to worry about.
If the repair of the runner as pictured is structurally sound, there is no need to repair it and nothing really to be gained by using screws. Man, that b/f likes screws, huh? If it is not structurally sound, again your best bet is epoxy.
I would not worry about a creaking rocking chair. In my opinion, that is part of the charm of an antique. However, if this isn''t your concept of a relaxing rocking experience, you will have to check all the joints to identify which ones are loose and in need of re gluing or repair.
Good luck and I think that it is great that you are jumping in on the chair repair project