The only measure of "integrity" I identify with sports is the capacity of the sport to entertain me. If I am entertained, the sport is doing what it is supposed to do.  
Generally, I am less apt to be entertained by sports that I think are farcical from a competetive standpoint.   This is one of the reasons I am not entertained by baseball since we are led to believe that teams with a $200M payroll are on the same competitive playing field as those with a $40M payroll. 
Despite stuckum on towels, heated footballs, deflated footballs, etc., I still view the NFL as a league that offers fair competition among its teams.  And since the sport is fun to watch -- I'll watch. The fact that fans from Indiana, Miami or Seattle might think Tom Brady is a cheater means nothing to me.  The fact that this might impact his "legacy" is irrelevant to me.  It's an entertainment industry and football is a game. Tom Brady will learn to live just fine with his deflated legacy. If he can't, then he has deeper issues. 
Is there a single person in Patriot Nation who will not root for the Pats to the exact same degree next year as it did this past year?  I doubt there is even one fan who has handed in his/her Pats Fan Card.  The integrity bar has been met as far as I am concerned -- they will entertain me.  That's all that matters. 
I am not sure there has ever been more ado about nothing.