The problem isn't needing to change card partners, it's not having (not having negotiated?) a 1+ year period where already-issued cards will continue to work. Having a year or more to gradually change cards the next time I buy something from each merchant would be a minor inconvenience. In the best case, it would be 2+ years and by then, it would hardly be a concern at all.
Privacy.com's mistake wasn't needing to change issuers; as you said, that's unavoidable. Rather, it was forcing customers to cut over existing cards in 3 weeks, then lying to customers by pretending this was an upgrade rather than something required by their issuer.