My suspicion is that Kensil, Vincent, Goodell, Gardi, Pash, and the rest of the top executives at the NFL league office viewed this as an open and shut case from the beginning and were more concerned with generating a satisfying PR resolution in the face of media and fan bloodlust, owner demands, internal grudges against the Patriots, and Goodell's ego. As soon as Kensil and Vincent saw the first ball that the Colts measured was beneath 12.5 PSI, because nobody in the league office understood the ideal gas law and they were already suspicious and resentful toward the Patriots, the entire aftermath with Wells and Pash and Vincent was a long exercise in confirmation bias and cognitive dissonance. Because the league office knew the Patriots had done it, Brady and Belichick maintaining innocence just embarrassed and infuriated them and they felt they needed to crack down. The misleading and falsified media leaks, the harassment of Patriots' employees to try to get them to flip, the one-sided advocacy piece that came out of the "independent" Wells investigation, the absurdly harsh punishment that exceeded allowances from player policies... these were the acts of people who were willing to circumvent due process and come up with any means possible to negotiate an admission of guilt from Brady in order to avoid being shown up by a player in a disciplinary decision yet again. Because they had already predetermined the outcome, the existence of a plausible explanation of weather effects being responsible for depressurization of the footballs only infuriated them because it gave Brady an excuse to maintain innocence rather than giving them pause about continuing the witch hint.
And I am pretty confident that this is the context for why Goodell arranged to hear the decision: if this had just gone to a neutral arbitrator, Goodell would have suffered the risk of losing his PR edge by his discipline being reduced or overturned without also getting the admission of guilt from Brady as part of the bargaining process. Goodell felt he needed maximum negotiating power to get Brady to fold and negotiate an admission of guilt in exchange for a reduction of suspension, and he was more concerned with using the arbitration powers as a bargaining chip to get that admission of guilt than with following through with due process by providing Brady with an independent arbitrator. It was just the next logical step after hiring an "independent" investigator to advocate against Brady and leaking fake information to make Brady look guilty to the public and creating an overly harsh initial punishment to create a strong negotiation position. It's not necessarily just that Goodell didn't trust a neutral arbitrator to uphold his ruling, though I imagine that was part of it. Rather, the primary issue is that Goodell has been far more interested in showing off his sheriff's badge and getting the good PR of cracking down on a cheater and getting him to admit to guilt than he has been interested in determining whether or not any cheating happened in the first place and what a fair punishment would be if cheating had occurred.