I have heard several;
Does double jeopardy apply? Article I, Section 3 states (in part); "Judgment in cases of impeachment shall not extend further than to removal from office, and disqualification to hold and enjoy any office of honor, trust or profit under the United States: but the party convicted shall nevertheless be liable and subject to indictment, trial, judgment and punishment, according to law." Remember, Trump was impeached for actions surrounding J6. The "experts" I have heard and read point out that a Senate conviction does not preclude another trial but the Constitution doesn't address if an acquittal precludes further legal action. Their position is, since there is nothing specific, we should look to normal trials for other matters. In those, an acquittal on a charge precludes further prosecution because of the double jeopardy prohibition. I think this is, by far, the weakest question.
Must "due process" be followed? Since a freedom/right is being taken, the due process provision (also in the 14th Amendment) must be followed. That would require a indictment and trial. A court, or attorneys, do not have the authority to conduct criminal investigations. The use of the word "criminal" comes from the next issue that has been offered.
Must someone be charged, tried and convicted before they can be disqualified? Section 5 of the 14th Amendment states: "The Congress shall have power to enforce, by appropriate legislation, the provisions of this article." Congress has addressed, by appropriate legislation, what constitutes insurrection when they passed legislation making it part of the US Criminal Code. The position of some "experts" is that, since there is a clear definition of insurrection in the US Criminal Code, a conviction is required before someone can be disqualified.
Does a state court have the authority to disqualify someone? Article III, Section 2 states (in part): " In all cases affecting ambassadors, other public ministers and consuls,
and those in which a state shall be party, the Supreme Court shall have original jurisdiction." Their contention is that since the Constitution give the Supreme Court original jurisdiction (because a state is clearly a party), they are the ones that must make the determination.
Is the President an "officer of the United States"? Article II, Section 2 (Presidential Power) states: "... and by and with the advice and consent of the Senate, shall
appoint ambassadors, other public ministers and consuls, judges of the Supreme Court,
and all other officers of the United States, whose appointments are not herein otherwise provided for,..." Their position is that "officers of the United States" are appointed, not elected. They feel the use of the term "appoint", a partial list of specific officers, and the phrase "and all other officers" make it clear the President is not an officer. They support that position further by pointing out nowhere in the Constitution is the President referred to as "Chief Executive". Article II, Section 1 only states; "The executive power shall be vested in a President of the United States of America."
There may be others, but these are the issues I have heard.