I think digital signatures and third party verification are an incredibly useful feature. The ability to prove you received some data from some third party lets you prove things about yourself, and enables better data privacy long-term, especially when you have selective disclosure when combined with zero knowledge proofs. See: https://www.andrewclu.com/sign-everything -- the ability to make all your data self-sovereign and selectively prove data to the outside world (i.e. prove I'm over 18 without showing my whole passport) can be extremely beneficial, especially as we move towards a world of AI generated content where provenant proofs can prove content origin to third parties. You're right that post quantum signature research is still in progress, but I suspect that until post-quantum supremacy, it's still useful (and by then I hope we'll have fast and small post quantum signature schemes).
EU's digital signatures let you do this for your IDs and https://www.openpassport.app/ lets you do this for any country passport, but imagine you could do this for all your social media data, personal info, and login details. we could have full selective privacy online, but only if everyone uses digital signatures instead of HMACs.