They *could* figure out a better system, but there's no practical way to keep track under the current system.
The ballots are filled out and then placed in an envelope signed by the voter. Once the signature is verified against the pre-submitted affadavit, the envelope is discarded. The ballot is then treated as an anonymous one. The voter does not sign the actual ballot.
Anyway, all someone has to do if questioned is say "I didn't use voter tickets to see '_____ show'. I went with a friend." So you have no official record of them seeing a show...but they say they did, and you are set up to take their word for it.
By the way, plenty of voters don't pay for League membership. SDC, Equity, etc etc etc have designated voters who don't spend $3500 to qualify. But they do donate a sh*t ton of their time serving in elected office. And frankly, these are the people you do not want to exclude, especially since the Tonys kicked 1/8 of the voters (a.k.a. the press) to the curb.
Such a fuss about Bette Midler sitting on a couch! As if that makes acting easier!
Tom Conti WON the 1979 Tony Award for laying motionless in a hospital bed in WHOSE LIFE IS IT, ANYWAY? And Mary Tyler Moore was given a special Tony for playing a distaff version of the role the following year.
That Midler's material is seen as slight is another matter.
I, personally, think sitting down makes acting much easier. In fact, I find everything easier sitting down. I had a 30 minute conversation standing in a parking lot the other day that almost killed me. It was difficult for me to concentrate on what I was trying to convey because my feet hurt, and I could barely walk the next day.
When I saw Lily Tomlin in Search for Signs for Intelligent Life in the Universe standing and moving around for 2 plus hours, I remember commenting to my friend -"That would have been SO much easier on her if she could have done the whole show from her couch."
Now lying down in a bed motionless for two hours is a completely different story. I toss and turn all night.