First, I don’t think we can take away any traditions from the Founding Fathers.  The penitentiary system came about after that, and the default penalty for any felony was death, so you wouldn’t really have any convicted felons running for federal office.  Second, I think it was implicit in the thoughts of the drafters that only the finest men in the community would be elected to federal office.
That aside, members of Congress are privileged from arrest during the session except for “treason, felony, or breach of the peace.”  I don’t have my federalist papers handy, but it seems that the first two being punishable by death and the third constituting an imminent harm were the reasons for those limited exceptions.
But under the hypothetical, the electoral college has voted for a person to become POTUS, knowing full well that the individual is under a sentence of incarceration under a state sentence.  There would be the constitutional quandary.
You would agree, I believe, that nothing would prevent that person from becoming POTUS.  I guess one could argue under the 25th Amendment that he is “unable” to discharge his duties because of his state incarceration, but a counterargument could be made that he is ready, able and willing but for the state interference in the federal constitutional process that made him president.
And, the founders were largely correct in that the people are not going to vote for someone, either as POTUS or a congressman, if they are in prison for crimes.  They are manifestly unfit for office.  But if under the hypo, people DO in fact vote for that person, then it seems to me that a higher order of democracy has spoken and one state should not be able to frustrate that.
Imagine a wholly corrupt deep red state (think Huey Long’s Louisiana type corruption) that trumped up charges against Biden, Harris, Schumer, et al. and convicted them and sentenced them to prison.  If the people of New York say, “Nope, we still want Schumer as our Senator” and the people of the nation say “We still want Biden as President” should one single state get a veto over a federal constitutional function?
However, like all of the what-if hypothetical constitutional crises, it has never been tested because of the extreme unlikelihood of it happening.  If it does, then the constitutional has failed on its own assumptions.