I think the clock is simply symbolic of a new game set up. This chess game has ended. Now we’re playing Reversi (or something) so everyone’s score (or clock) is reset to zero.
Also stopping the clock is just useless logistically anyway because Jared Gilmore isn’t going to quit aging. If they’d killed Henry, they could have stopped the clock. But he’s alive and well and the kid’s not out of a job so the clock must still be moving.