Vote-by-mail ballots are fed through a computer tabulation system once received. In some states, especially those that are entirely vote-by-mail like OR and WA, large centralized tabulators are used that read multiple ballots per second from a stack feeder. In most states though it's very common for mail-in ballots to be fed through the same precinct tabulators as in-person votes, just at the election administrator's office instead of in a precinct location. This "mail-in precinct" approach has the advantage of keeping the process very consistent across voting methods, although it tends to require more staff since precinct tabulators are not built to quickly feed large numbers of ballots.

Remember that hand-counting of ballots has been unusual in the US for quite some time. Some states process their mail-in ballots through large central tabulators even though they have only a small portion of mail-in votes, simply because they still have the central tabulators from before they switched to precinct tabulation, back when they used to drive the ballot boxes from every precinct and have staff re-stack the ballots and feed them to a tabulator after close.

Ballot secrecy is unlikely to change. It was widely adopted in the US as a direct result of the fact that non-secret ballots facilitated purchased and forced votes, since the payer could verify that the voter cast the ballot they were supposed to. This is not a theoretical problem but one that was widespread in the 19th century.

Something like you describe is already in available in many states, though, at the pollbook level rather than the tabulation level: in many states you can obtain a record online of whether or not you were issued a ballot and, if you were issued the ballot by mail, whether or not it was received back. The ballot is 'severed' from this record system (usually by physical means like dual-envelopes or even passing through a slot in a wall) before tabulation so that your voting choices cannot be proven after the fact to facilitate bribery or intimidation.