Its very much implied that when we beat the Shade of Nidhogg and riped out the eyes the spirit of Nidhogg left for good. The eyes are likely simply Aetheric batteries. Even if his rancor was present, I don't think it would be enough to fill the role of believer to sustain the primals form.

Simply put there is a lot of reasons why it would make sense to be Rhalgr and very few to have Shinryu make sense. If Shinryu does turn up it will likely be in the Bend of Time/Interdimensional Rift. Infact we might find out that Shinryu is a dragon from wherever Midgardsormr came from.