Lindsey is not a judge, he is just one of the Senators, acting as a jury.
“In impeachment proceedings, the House of Representatives charges an official of the federal government by approving, by majority vote, articles of impeachment. A committee of representatives, called “managers,” acts as prosecutors before the Senate. The Senate sits as a High Court of Impeachment in which senators consider evidence, hear witnesses, and vote to acquit or convict the impeached official. In the case of presidential impeachment trials, the chief justice of the United States presides. The Constitution requires a two-thirds vote of the Senate to convict, and the penalty for an impeached official upon conviction is removal from office. In some cases, the Senate has also disqualified such officials from holding public offices in the future. There is no appeal. “
https://www.cop.senate.gov/artandhistory/history/common/briefing/Senate_Impeachment_Role.htm
I thought CJ of SCOTUS oversees the proceedings.
I may have it wrong.