Ideally those requests would also include a "deleted after" date. If such a thing was implemented and trustworthy (kind of hard, admittedly), it would make me a fan of an EU-wide eID.
That would make definite sense and should be straightforward. Instead of sharing your data, you would generate a certificate with an expiration period with which they can access your personal data stored in the system. Ideally this would be short lived and require periodic regeneration.