About the only sure way to do what you want done is to have a professional shop do the stripping. It is almost impossible to hand strip oak, for the reason you noted.

My best advice, if you try this yourself, would be to use a stiff bristled nylon brush(like in the link) after letting the stripped sit and soak.

Be aware, this means the detritus and chunks will be flung all over the area and make a HUGE mess.

Hate to be Debbie Downer but If you are a newbie at refinishing - you couldnt have picked a more difficult project due to the intricate nature of a staircase railing ..... as opposed to something flat like table or door. I would see if I could negotiate something with w professional - ie if I was to do the grunt work of stripping off most of the old poly maybe he would give a discount for doing the final futzing with stripping/sanding and applying the finish.

If you arent a newbie, if you know how tricky it is to get into all the tiny nooks and crannies and can get your hands on or canfashion some tools for getting in to the small places then what the heck, go for it - it can be a labor of love and so it takes you a while - a post every weekend is a realistic goal. Of course it would be better to do this before the treads are refinished - that goop has a way of getting everywhere

If it was the old hide glue you could disassemble it to refinish but I dont know if that's possible with modern glues.

Wouldn't it be vastly easier to simply remove the old railing, spindles and posts and purchase new everything? Aside from the huge length of working time, stripper is expensive. Or perhaps throw all removed parts into a commercial stripper tank?

I've never seen stripped wood take a nice stain unless sanded first.