Maybe you need to brush up on separation of federal powers. Legislation, spending bills, all manner of BS that flows out of DC starts in the Congress.
Conservatives take both Houses and control the committees and legislation leverage of powers, Obama could do only executive orders, his powers are actually quite limited. He only signs bills or vetoes them.
I'd much rather gain the prize of Congress than see that idiot Biden as President with a democrat controlled Congress if the usurper Obama is run out of office on a birther issue, which is highly unlikely even with a GOP controlled Congress.
Winning Congress does not require giving up on the eligibility issue.