Earlier quoted context omitted.
I wonder if the Senate could pass a rule that allows them to choose when to take a secret vote. I don't know of anything in the Constitution that prohibits such.
The Senate routinely takes voice votes for which no rollcall division is recorded. But there is generally little incentive to make votes private because in general legislators want to engage in credible credit-claiming (including about margins of passage and rates of bipartisanship) and parties want to preserve the whip as well.
You don't need a rollcall to know why you lost if not even enough people were in favour of your proposition to even force the actual count.
The intended reform would be that the rollcall would happen, yet the details (beyond a total Yes/No count) would not be released, presumably even to the voters themselves. This is a problem because it's hard to maintain trust in such a system.
For popular elections the scale is such that election monitors on behalf of interested parties can easily see a bunch of administrators correctly tallying votes and not focus in on the fact that, hey, voter #283204349 picked Jim Smith for Grand Leader -- however when you're talking about exactly 100 senators that's surely much harder to get right.