As I see it, the "failure to qualify" provision of the 20th Amendment can't be based on some arbitrary decision in a political or legal process that isn't already in place under the Constitution and/or Federal law. In other words, Congress doesn't have the authority to simply decide on January 6th (for example) that Joe Biden doesn't qualify to be the President because someone in Congress (or the Vice President, acting as presiding officer over the joint session) says so. Rather, the "failure to qualify" can only be established AFTER the Article II and 12th Amendment process has played out. One scenario that might apply here would be where the electoral vote in December produced no majority for any presidential candidate (suppose the Libertarian or Green candidate won a handful of swing states, for example), the presidential election was passed to the House under the 12th Amendment, and at the January 6th session of Congress the House vote for President ended in a tie. That is certainly not outside the realm of possibility when you have an even number of states, and/or you have state House delegations that are split evenly between Democrat and Republican representatives.
Since the 12th amendment did away with separate voting for President and Vice President, and the 20th amendment contains language for what should happen if either the President-Elect, Vice President-Elect, or both, should "fail to qualify," I don't see how your scenario might result in one, but not the other, failing to qualify. Yet the language is there in the Constitution for it to happen.
My larger concern is about the nature of the parties. When an ambiguity like this exists in the Constitution and the law, it is the nature of the Democrats to push it as hard as possible to drive the "untested waters" to the outcome most desirable to them.
When Republicans are in the same situation, they talk themselves out of doing anything for fear of the "worst case scenario," or they make the most burdensome arguments as reason for not testing the waters themselves.
This is why the Democrats always win and Republicans always lose. Democrats always put the ball in play to see what happens in an uncertain situation, and Republicans refuse to play because they prematurely convinced themselves that they will lose.
-PJ