(c)(1) The Board shall be composed of 7 members who shall be appointed as follows:
(A) 3 members shall be appointed by the President, by and with the advice and consent of the Senate. Not more than 2 of such members shall be from the same political party.
(B) 2 members (each member from a different political party) shall be appointed by the President pro tempore of the Senate with the advice of the Chairman and the Ranking Minority Member of the Senate Committee on Finance.
(C) 2 members (each member from a different political party) shall be appointed by the Speaker of the House of Representatives, with the advice of the Chairman and the Ranking Minority Member of the House Committee on Ways and Means.
(2) The members shall be chosen on the basis of their integrity, impartiality, and good judgment, and shall be individuals who are, by reason of their education, experience, and attainments, exceptionally qualified to perform the duties of members of the Board.
The President names three one of which comes from the Republican side. Boehner and Reid each pick one and gets it with "the advice" of a ranking member of the other party.
The President gets his Republican name from the ranking Republican, in this case McConnell.
We nominate our guys and they nominate theirs.
When Bush was President we still got to "pick" our representative. Do you really want the Republicans picking our guy? Well they don't want us picking theirs either. This system has been going on for decades and works just fine.