Building a better voting system is actually very simple. You can't ask people to trust the computers because computers can easily be corrupted by bad actors. However, computers can still be used to make voting a modern experience and one that people can trust more than even the best of out current systems.

This system would be implemented by the government but verified by everyone. What the government would do is ask everyone to email in their vote. Each email would include the voters name, address, tax ID, and of course who they are voting for. If they mess up the email format then an AI can easily ask them to clarify their information. The government would then add their vote to a CSV file but it would hash their personal information first since the CSV will be published for all to see. The voter would then receive their hash in a reply, though the hashing algorithm would also be open source so people could generate their hash if they want. This is important because this is what enables them or anyone who knows their name and address to see who their vote was for. If people don't want their friends to be able to see who they voted for they could ask to have their tax ID included in the hashed value. Since most people don't know their friends tax ID this should be enough privacy for anyone. Once voting closes the government will email the CSV to everyone so they can verify that their vote was counted correctly.

This system would be prone to corrupt behaviour on the part of the government since it's up to them to verify the tax ID's and ensure that no fake tax IDs are counted. However this is easily countered with statistical verification. Which would be done by 3rd parties that people could register their votes with. If the government provided data differs in a significant way from their private data then they can sound the alarm. Now obviously the government could attempt to corrupt the 3rd party but since there can be as many 3rd parties as people want then it quickly becomes too expensive to do so.