(1) The scale of integrity (from perfect to abysmal) and the continuum of success (from 15-0 down to 4-8) are not correlated. There is no significant causation. Sure, cheating can get you a win or two in the short term, but in the long term it can/may cost you more. There is absolutely nothing wrong with wanting our program and its leaders to be ethical, mature, respectful and respectable. We can have all that and win championships, too. Just gotta get the right leaders.
(2) Fan bases are vast, amorphous things. Don't be fooled by VN.com; the full Volunteer Nation is huge, massive, you can't begin to know one-tenth of one percent of all of it. Yes, you can characterize the fan base as a whole, but you must use very general statements; otherwise, you lose coherence with the entirety.
(3) What's happening right now between the UT administration (including our inept AD), the booster community (factionalized as it seems to be) and the part of the fan base that is engaged (probably well under 5%) is important. It may be program-defining for the next decade. It is chaotic and messy and, in places, ugly. But that doesn't mean it might not, in the medium and long term, be the best path forward, the most beneficial option.
(4) The national media can reasonably be ignored, for now. They have not yet shown that they see any of the real currents flowing around the head coach search. All they see are wave tops: Shicano offered, vocal outcry, Schiano rejected, Brohm offered, Brohm declined...that's all they see. They are blind to the behind-the-scenes battle for control among the boosters. They never saw in any detail the hopes of the fan base, the expectation for a Patterson-Peterson-Gruden level outcome, so they do not "get" the disappointment of the offers made. They are part of the "forgive any sleaze after a bit of time" mentality brought to us originally by Hollywood, so don't understand why values matter when a Schiano or Kiffin is mooted as an option. It is almost astounding how many ways they don't see the full picture. Ignore them; one day, they'll catch up with some of it.
So back to the OP: is Tennessee tarnished? Not really. We are in the middle of a power struggle for control of the athletic programs (and possibly much more), but we are not tarnished. Eventually, it will all be sorted. Hopefully, we'll be far better off for it.
And the fans? We're just being fans, with every possible element you can imagine represented somewhere among the millions of us. Including those (the majority) who have very little idea any of this is going on.
Go Vols! To a brighter future!