In particular - they are much less likely to always blindly follow party lines, or corporate wishes, and I think there's value there.
How would you feel about a tweaked system? All votes are recorded, but the voting record is entirely withheld until 3 months before the next election for that seat.
The public is made aware of the voting record when it matters: During the next election. But during the term, they are much less susceptible to corporate donors and lobbyists, or pressure from members of their own party.
I want the politician I vote for to more or less follow the party line on important votes. That's why I voted for her.
I also think this leads to a preponderance of "division" politics - where the goal is to find a wedge issue that forces people to pick a side.
It's poor governance all around.
(Obviously disclosing your cryptographic vote receipt to a donor would be forbidden.)
I think explicit quid pro quo is far less impactful/prevalent then pro-corporate donors simply looking at the pool of campaigners and putting weight behind candidates that look like they'll vote how the donors want. All that requires is for the donors to be able to at least partially predict future behavior from candidates' current appearance. Secret votes won't change anything on that view.
Also, small dollar donors are driving polarization of US politics, so I'm not sure big donors are really the problem.
1) Politicians decide to pull a fast on corrupting interests and start voting in the interests of society instead of focusing primarily on what benefits themselves and special interests.
2) Politicians decide to pull a fast one on society and start defacto selling influence more than ever, given they can simply publicly claim they picked the "right" side of any given bill.
Just like on a jury for a court case, people are randomly selected from the population to sit on the legislature. They naturally vote their conscience, and when their term is up, they go back to their normal life. Because compliance with a bribers' instructions is unobservable, bribery is not incented.
The beauty of it is that you don't need to hold anyone "accountable" because there is no election, only random selection. Of course, the devil is in the details. Should prisoners be eligible for selection, or the illiterate?
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.
A constitutional amendment would nuke the current limitations, but it's impossible in practice for us to get there. On the other hand, as we've seen as of late, no decision by the Supreme Court is set in stone, so we could, in theory, get Citizens rolled back and then some. That also seems unlikely in the short term.
Not that this is in any way likely to happen, of course.
Notably, the Constitution does not grant the court the power of judicial review. The court granted itself that right. So in a battle between precedent and the text of the Constitution, I hope the latter would win.
However, another political reform I've toyed with in my head is secret voting in the House and the Senate.
Who cares if Raytheon or Disney "owns" a given representative if those corporate sponsors can't actually be sure that they're getting the votes they paid for. It would also neutralize a lot of the petty party line votes and manipulation. Every issue would ostensibly be decided based on the representatives' actual value system.
Obviously the downside would be that the public at large would be unable to hold their representative accountable for a particular vote, but that seems like the lesser concern compared to the very obvious corporate vote buying and party-line bullshit that happens today.