If no rose experts answer what I would do at this stage is to cut back the healthy branches a little way to an outward facing bud, and cut the weak or dead growth right back. If this delays flowering a little it won't really matter. Therte's still time to give them a reed with rose food.

Lack of pruning will not, in itself, cause weak growth. Any growth from the unpruned twigs will be thinner than normal, the flowers from that will be smaller. On the other hand, new canes growing from the base are likely to be thicker than usual, since they can access more of the food that the bush stored last summer. The main problem is that the bush is turning into a thicket that will be hard to care for properly, and be even harder to prune next spring.