Correct, and the parties choose the candidates unless one runs as an Independent. Regardless, the qualifications are determined by the Constitution and until now it was assumed that no one ineligible would run. The opposition and the media were expected to assure this. This time it failed primarily because of the media and the fear o it.
A challenge in court seems the only remedy and finding a court that will hear it, and give it a fair hearing, is exceedingly rare.
The constitution anticipates the possibility of an unqualified person obtaining the majority of electoral votes. The failure here, if there is one, is squarely a failure of Congress.
Same with the financial meltdown - Congress shall coin money and regulate the value thereof, etc. The condition of the economy is more a failure of Congress than of the executive.