The Succession Act simply skips ineligible people. For instance, a foreign-born person could well be Speaker of the House, but in the event that the President and Vice President were rendered unable to discharge the duties of the office, the Speaker would simply be skipped and the President pro tempore would be sworn in as President. The reason that Vice President is special is because of the 12th Amendment that specifically forbids anyone ineligible for President from being Vice President.
This is true. Henry Kissinger was not elibible to be POTUS, but became Sec. of State nevertheless. Sec. of State is pretty high on the food chain of presidential succession.