The fact is that a US Senator does not have to be, nor is required to by the Constitution, to be ‘impartial’. The requirements for being Senator are finite and written clearly. The responsibility any Senator owes is to the state which he represents and the nation as a whole. Partisanship, preferences, biases or proclivities not reaching the definition of impeachment are at a Senator’s discretion.
In truth, the public won’t decide crap about Cruz’ authority to vet; that is ALREADY given by his being a Senator. It is up to those who want Obama’s choice, likely a very bad one, to get enough Senate votes to overcome dissent.
What the author asserts is bullshit and it is a diversion and attempt at influencing this Senator’s standing in election for higher office as well as intimidate him to change his likely opinions and advice and consent on a duty he is already sworn and authorized to execute. The Constitution says nothing about all these specious suppositions of IFs, CANs or MIGHTs, and it is a waste of time to suffer the machinations of these fools any further than right now.
You can wishful think and hope that these bullshit claims, etc. have no play, but your influence is limited. Some people will continue to believe that Cruz is either outright ineligible (I’m in that group), or will discount his campaign because they fear the risk of litigation over the issue. Both of those will dilute Cruz’s share of the vote, and the ABA piece is, of course, an effort exactly along those lines. At the same time, the ABA piece probably has less effect than Trump beating the drum.
There are Senate (and House) ethical rules regarding conflicts of interest. So the general principle being asserted is legitimate. But those rules as presently written pertain mainly to advancing or passing legislation that benefits the legislator financially. (For the same reason, Presidents or candidates have put their assets into "blind" trusts to guard against the perception they may take action knowing it will benefit them personally.)
Though, as the article notes, Senate rules at present don't clearly address the situation Cruz would face in his role on the Judiciary Committee. So there would be no actual requirement for Cruz to limit his participation.
If there were an actual case involving Cruz within the federal courts that was pending or likely to come before the SCOTUS the pressure for Cruz to recuse himself, at least from his Committee role, might be stronger. But that isn't likely to be the situation.