The podcast you were listening to was discussing an internal caucus vote by the Republicans. The "Senate" didn't vote on a secret ballot for John Thune as majority leader, the Republican senators voted amongst themselves to appoint him to lead their voting bloc. And since they had enough votes to control the Senate, he became the majority leader.
That's a rule the Senate Republicans made themselves, they could just as easily decide to hold a public vote. But either way it'd be an informal vote and not an official Senate vote.
That said, the rules of the House and Senate do allow for closed sessions in which all proceedings, debates, and votes are behind closed doors. That's in line with Article 1 of the Constitution, which requires that houses of Congress keep a record of their proceedings, "excepting such Parts as may in their Judgment require Secrecy."
"Closing the doors" was once very common, but is now done extremely rarely, pretty much only for impeachments, treaty discussions, and nominations that touch upon sensitive personal subjects or issues of national security. Most recently the Senate went into a closed session during the 2020 nomination of Amy Coney Barrett to the Supreme Court, and before that one has to go back to the Senate ratification of the New START treaty in 2010, when Congress understandably did not want to debate U.S. nuclear weapon policy in public. The House hasn't closed the doors since debating surveillance of terror subjects in 2008.
