Yes, they are eligible. Which just means they aren't legally disqualified from office. They'd still have to win the election. There doesn't have to be legal disqualification against everyone we don't want to win.
I don't agree with your assertion. Why should the child born of a tourist or an illegal alien be considered eligible to run for President an NBC as required under the Constitution? I would challenge any such any such candidate in the courts. It is the reason why this issue needs to be adjudicated by SCOTUS to define exactly what is an NBC for the purposes of being President or Vice President. These are the only offices we are talking about, not all elective office.
"Citizen at birth" is not "natural born citizenship". Those are entirely different concepts. Many people are made citizens at birth by statute. That is what the statute did that retroactively made John McCain a U.S. citizen at birth, or the statute that makes persons born in Puerto Rico U.S. citizens at birth, or some provisions of 8 USC 1401, but those are naturalization statutes, and one can be naturalized at birth. It doesn't have to be done after birth, through a congressional "private bill", or through an application process.
For example, natural born citizens of Puerto Rico are not natural born citizens of the United States, eligible to be president, while it remains a protectorate or dependency. However, citizens of Puerto Rico have been naturalized by statute to be citizens of the United States at birth. Now if Puerto Rico were admitted as a state, its natural born citizens would then become natural born citizens of the United States, eligible to be president, if otherwise qualified.
No Supreme Court opinion has "defined" natural born citizenship for purposes of presidential eligibility. The cases cited were either dictum or concerned ordinary citizenship sufficient to vote or hold office, but not to serve as president.
The burden of proof of eligibility is on the candidate, not on one challenging eligibility, and he must be presumed to be ineligible unless or until he can produce the proof.