Hernandez and Franco both wore C's on the jerseys. Carter was also a co-captain with Keith and didn't wear a C. So this was way before Varitek.
Jeter and Wright made this all look silly by leading by example without the need for a C on their jerseys.
To each their own, some leaders like to be very vocal and have a need to let all know who the C is with the C patch on their jersey, which is fine, others just lead by example, which is also fine.
I think since Hank was not allowed to wear a C as a goalie and was likely the locker room leader and de-facto captain for three years until leaving for washington, the team probably decided they did not need anyone to wear a C just for the sake of having fans elevate one player over their teammates.
That said, 6 alternates seems totally silly as it seems akin to a participation award to be awarded an A now.