[peonnolm] Piece power in large chess

We consider large square board chess variants which are formed by simply increasing the number of rooks and bishops (and pawns).  For random positions (leaving unspecified what is "random"), what percentage of squares are attacked?  How does the percentage change as the board size increases?

The worry is that as the board size gets larger, because the relative portion of the board a single ranged piece can attack shrinks (curse of dimensionality), so the game will devolve into a boring game with relatively weak pieces.

What percentage of the board can be attacked in 1 move (the above was considering attacking in 0 moves)?  On an empty board, a rook can attack any square in 1 move, no matter the size of the board.  For large boards, the relativity sparsity of pieces might aid the ranged pieces.  Is being able to attack in 1 move useful?  The defender can often escape in 1 move as well.

If we need more powerful pieces, we could consider pieces that can fly or do hook moves or drops, though drops are similar to attacks in 1 move since you cannot simultaneously drop and capture.