If No, please provide reasons and suggestions.
Mutual recognition of registration in different jurisdictions still imposes the cost and paperwork of registering in each jurisdiction. Having the same baseline qualifications is a good starting point, but for a structural engineer to need to separately register (and pay registration) in every state that their clients are present is an impost on them that seems to be driven wholly by paperwork rather than providing any real benefit.
I suggest that an engineer can apply to be registered in a state by providing that they are on the NER (or another register), and that the state registration isn't an additional fee for the engineer.