The biggest problem with doing away with secret ballots is that it makes fraud verifiable. Let's say Mallory pays Alice $20 to vote for socialist candidate A. With a secret ballot, Alice can go into the voting booth, and cast a ballot for Constitutionalist Candidate B, step out of the voting booth and collect her $20, with Mallory being none the wiser.
I don’t know what the solution is but with secret ballots fraud can take place by adding or removing ballots from the count and nobody will ever know.
I think it might be easier to catch vote buying than secret ballot fraud.