You ask: Could the President recess appoint himself into the chief justice’s position during his own impeachment? Your query raises the same subject as:
[a] May the Speaker preside when debate on the House floor involves investigating, censuring, disciplining, and/or expelling the Speaker?; and,
[b] May the Vice President preside when debate on the Senate floor involves investigating, censuring, disciplining, and/or impeaching the Vice President.
These situations are sometimes put forward as examples of defects in our Constitution, but this critique fails. In each of these situations, the relevant house would just go into the committee of the whole, and that would displace the presiding officer, which would resolve the conflict at issue. The committee will report back to the full House or Senate. If the committee had a majority supporting the imposition of some punishment, then the same majority can act when the committee reports to the full house. If two-thirds were necessary to take action, the same scenario would apply. If a two-thirds majority were available in the committee of the whole, then that same two-thirds majority can act when the committee reports back to the full house. At that stage, where the merits had been fully determined in the committee, it would not matter who was the presiding officer or if he were conflicted.
The only defect would be if the original motion to put the proceedings into the committee of the whole led to an evenly divided House or Senate. I am not going to lose any sleep in regard to that potential minuscule problem. Let’s get real here. Consider the scenario involving an impeachment of the Vice President before the Senate. It is very unlikely that a Vice President will be impeached; it is also unlikely that an impeached Vice President will attempt to preside; and it is also unlikely that an impeached Vice President, prior to trial in the Senate, but who demands to preside in the Senate, will have the benefit of an evenly divided Senate.
Seth Barrett Tillman, Could the President Recess Appoint Himself into the Chief Justice’s Position During his own Impeachment?, New Reform Club (Nov. 18, 2019, 8:01 AM), <https://reformclub.blogspot.com/2019/11/could-president-recess-appoint-himself.html>;