Why shouldn't we update Killers for Captures?

[TheDarkKnightRise]

        // If ttMove is quiet, update move sorting heuristics on TT hit (~2 Elo)
        if (ttMove)
        {
            if (ttValue >= beta)
            {
                // Bonus for a quiet ttMove that fails high (~2 Elo)
                if (!ttCapture)
                    update_quiet_stats(pos, ss, ttMove, stat_bonus(depth));

                // Extra penalty for early quiet moves of the previous ply (~0 Elo on STC, ~2 Elo on LTC)
                if ((ss-1)->moveCount <= 2 && !priorCapture)
                    update_continuation_histories(ss-1, pos.piece_on(prevSq), prevSq, -stat_bonus(depth + 1));
            }
            // Penalty for a quiet ttMove that fails low (~1 Elo)
            else if (!ttCapture)
            {
                int penalty = -stat_bonus(depth);
                thisThread->mainHistory[us][from_to(ttMove)] << penalty;
                update_continuation_histories(ss, pos.moved_piece(ttMove), to_sq(ttMove), penalty);
            }
        }

[xu-shawn]

This part of the code has little relationship with killer moves. But to answer your question, killer moves prioritize certain quiet moves in move ordering. Therefore, in Stockfish's move ordering scheme they come after good captures and before quiet moves. It makes little sense to store capture as killer as in doing so one would likely push it towards the back of the move order, which is suboptimal.

[xu-shawn]

Note that there is nothing you should or shouldn't do. Every idea gets tested on Fishtest and Elo is the only thing that counts.