"An advisory board is the president's way to get confidential advice, and if you have a member of Congress on the board, are they functioning as a member of the president's board or as a member of Congress?" asks a former White House lawyer. In any conflict with Congress, a president could never claim privilege over the advice he received if a member of Congress were part of the board giving him the advice.
"From a separation-of-powers view," says another former White House attorney, "a president wants command and control over his executive officers, and he wants input to come either from a member of the executive branch or a person reporting only to the executive branch."
That's why presidential boards bar members of Congress. But what about the other senior-executive-branch boards Bauer mentioned? Many of them don't include lawmakers, either. Take the Defense Policy Advisory Board, which advises the Secretary of Defense. There are no members of Congress on it, for the same reasons they don't serve on presidential boards. Would the board member be working for the Pentagon or for Congress? Besides, the Defense Department, like the president, gets plenty of advice, solicited and unsolicited, from Congress.
Finally, the Constitution forbids anyone from holding offices in both the legislative and executive branches at the same time. Article I, Section 6, known as the Incompatibility Clause, says "no person holding any office under the United States shall be a member of either House during his continuance in office." An unpaid board position for Sestak would probably not violate the clause -- that is perhaps why Bauer emphasized that the positions were unpaid -- but the principle of separation of powers is clear.
The bottom line is Sestak, as a member of Congress, could not have served on most, if not all, of the boards the White House claims it considered for him. Did the politically savvy Obama team really not know that? Or is the White House not telling us something?