Ahh, but it does. It undulates, up & down, as well as into, through & out of, the spiral arms. We, at present, just happen to be 'in the clear' at the moment.
A few years ago, 2 young American Uni students did some calcs. on 'Solar' recession/movement, & found that at about 60/65,000,000 years ago, 'WE' were in close proximity, (if not actually in it), to an arm. It could have been a Supernova in the spiral arm close to 'US' that caused the demise of the dinosaurs..... ie; radiation etc etc from it.
Ahhh, theories. They make you think, don't they?
L.