"Is it really so unlikely that closing just a few weeks before the Tonys will ruin their chances?"
Unfortunately, yes. At the end of the day, Broadway is a business. Even with stellar performances, the show has never found an audience. This speaks to voters that regardless of how good it is, there is little to no interest in this property. (If it tours, who would come?). Also, the reviews were lukewarm. If it were to win, there is no money to be made for producers and no award winning performance for audiences to see.
It hurts Kelly significantly.
"The sexual energy between the mother and son really concerns me!"-random woman behind me at Next to Normal
"I want to meet him after and bang him!"-random woman who exposed her breasts at Rock of Ages, referring to James Carpinello