There's a FAQ on the same page that talks about how the modern nominees are limited to 5:
https://www.profootballhof.com/hall-of-famers/selection-process-faq/
"In advance of the Hall of Fame selection meeting, the Selectors are provided detailed biographies on each of the 19 Finalists. At the annual meeting, each Finalist is thoroughly discussed by the committee before a series of reduction votes are taken. First, the Coach/Contributor and Seniors Finalists are discussed and voted on for election. They must receive the same 80% affirmative vote as the Modern-Era Player Finalists. Next, there is a vote that reduces the Modern-Era Player Finalists list from 15 to 10. ***Following that, a vote is taken to reduce the list from 10 to 5 names.*** The five remaining Modern-Era Player Finalists for Hall of Fame election are voted on individually (yes or no) for membership."
*** emphasis mine
So Gates almost certainly made the first cut from 15 to 10, but didn't survive the second cut from 10 to 5, where Hester, Peppers, Freeney, Johnson, and Willis all made it, and they all obviously reached the threshold in the last vote.
I'd assume he gets in next year, but in addition to his fellow finalists Torry Holt, Reggie Wayne, and Jared Allen, there are first timers on the ballot next year such as Eli Manning, Luke Kuechly, and Terrell Suggs. It's going to be crowded getting down to that last group of 5.
Michael