Biden ran as the VP candidate in ‘08. He was elected. It doesn’t matter WHO chose him to be that candidate.
Biden is first in the line of succession. Obama’s being declared ineligible and a usurper would fit in the category of ‘unable or unfit’ to serve. The constitutionally established succession would kick in. There is no question about Biden’s eligibility to serve as VPOTUS under the Constitution. If, on the very long shot, Obama were removed from Office, Biden would take his place constitutionally.
He was not elected separately. He came in the package and the package is denied. If he for some unlawful reason did become President he would be a laughing stock. It would be like putting two stakes into the heart of the Democrats.