I agree that convincing the public is a separate issue. But on a technical level, these problems are solved. See e.g. this modern implementation: https://github.com/vocdoni/davinci-paper This is a pretty recent one, but the mathematical foundation dates back to the late 80s.
There have also been earlier implementations like this one: https://github.com/benadida/helios (corresponding paper https://www.usenix.org/legacy/event/sec08/tech/full_papers/a...), although it still lacked coercion resistance. But it already uses zero knowledge proofs, which is at the core of anonymous yet verifiable systems.