How does government with its control over banks not already knows what happens in relation to all those bank accounts? Would switching to Govcoin really change anything materially in that respect?

The people that do not wish for transparent ledgers, will have the choice to switch to private cloaked ledgers. The demand for such ledgers will be sufficient for them to seek their use even if banned by law. This notion is enforced if the Government would use public ledgers and total control to extract punitive levels of taxes. No better incentive to escape the system than punitive taxes.