Privity is well settled by House of Lords, Canada, and here. Leave is needed to re-open an earlier decision of the Court of Appeal to review doctrines suited to an earlier age but which work injustice in contemporary situations. To be exercised sparingly. If change is contemplated, a substitute doctine sufficiently precise must be at hand. A right for the third party is to create a legal relationship which the doctrines of our legal system are not able to define.
Possible responses are varied and complex.
Recognition of this would require new source or rights and obligations foreign to our common law system. So called exceptions are really the application of other principles - trust and agency. No true exception to the doctrine of privity. Attempts to introduce it would raise as many problems as it would solve. Action by promisee and specific performance should be sufficient.