But with bishops, rooks, and queens having such "long range" influence, it is frequently difficult for me to account for them when they are outside my "mental window", which is at best about 3x3 squares (the influence of a knight).
You're definitely not alone there.. I have problems with that too.. It's easy to zoom in and overlook outside influences.
I can't say for certain, but it really seems to me that part of the problem with remembering the calculation once you step back to look at something else might have to do with how you connect everything.
For instance, the sequence 1.e4, e5 2.Nf3, Nc6 is probably fairly easy for you to visualize and remember. The key there in my opinion, is that you're very familiar with the starting position. You have a frame of reference to relate the movement towards.
As things progress the pieces can tend to become a half-hazard jumble. It frequently isn't very similar to the starting position at all. The mind needs another reference to relate everything towards. This pattern recognition starts to happen naturally with a lot of frequent repetition.. (f3 knight being pinned by the a bishop on g4, King on g8 covered by pawns on f7, g7, and h7 etc...) However getting this pattern recognition by rote repetition isn't very efficient, and it's usually only related to a section of the board that tends to repeat.
I'm a big proponent of studying the pieces in isolation. Drilling into your head how any one piece can move to any other square in the shortest amount of time. Combined with different ways of visualizing the moves, particularly the knight. (For instance, most people view the knight as an L. Some people see the move as a move up or to the side from a diagonal adjacent to the knight.. Still others view it as a square of the opposite color on the outside of a 3x3 square.) It may seem silly to practice visualizing piece movements in different ways since they all amount to the same move, but I've found it cements the imagery of the chessboard into your head. Furthermore it allows you to intuit rather than calculate how all the pieces are connected.
Combined with a framework of the board, so that you automatically know where b5 is without thinking, you can see it. A person should be able to calculate sequences more accurately. As to considering how all the squares relate to each other, as I mentioned earlier I visualize from the inside out.. The center acts as my point of reference.. That's my preference but so long as you are able to relate all the squares to a point of reference you should be able to place them without thinking.
I'm certainly no genius nor master chess player, so take what I say with a grain of salt, but this is what I've found to be true for myself in terms of board visualization and calculation.
Can intelligent person suck at chess, forever?
As an intelligent person you should know that before you played chess you had the potential to be a GM or suck at chess.
In fact you existed both as a GM & a Patzer until you measured your playing ability - then you found out you re on the "I suck" end of the scale.
As many people have said , stop playing so much fast chess. join a couple of groups that play Vote Chess matches and listen to the thought process of the higher rated players. Tactics trainer is a good idea aswell